FANO VARIETIES of LOW-DEGREE SMOOTH HYPERSURFACES and UNIRATIONALITY Submitted to the Department of Mathematics in Partial Fulfi

FANO VARIETIES OF LOW-DEGREE SMOOTH HYPERSURFACES AND UNIRATIONALITY

AUTHOR: ALEX WALDRON

ADVISOR: JOE HARRIS

submitted to the Department of Mathematics in partial fulﬁllment of the requirements for the degree of Bachelor of Arts with Honors.

Harvard University Cambridge, Massachusetts March 31, 2008 1

1Email: [email protected] [email protected] Contents Introduction. i 0.1. Dimension of Fano varieties i 0.2. The low-degree limit ii 0.3. Application to unirationality in low degree iii 0.4. Further work iii 0.5. Incidence correspondences iv

1. Planes on Hypersurfaces in General. 1 1.1. Deﬁnition of the Fano variety 1 1.2. Estimating the dimension of Fk(X) 2 1.3. Proofs by example. 5 1.4. Fk(X) has the estimated dimension for general X 7

2. Fano Varieties in the Low-Degree Limit. 11 2.1. Notation and terminology 11 2.2. A ﬁrst result 11 2.3. Constructions 13 2.4. Explicit description of the ﬁbers of the residual varieties 17 2.5. Numbers 19 2.6. Low-degree smooth hypersurfaces do not have too many k-planes 20

3. Unirationality of Smooth Hypersurfaces of Low Degree. 27 3.1. Preparatory results, and combs 28 3.2. Statement of results concerning unirationality 30 3.3. Constructions 31 3.4. Proof of unirationality of smooth low-degree hypersurfaces 33 Acknowledgements. 36 References 37 Introduction. n Given a projective variety X ⊂ P and an integer k > 0, we ask a classical question: does X contain a projective linear space of dimension k? As it stands, this question invites the following explicit line of attack. Let {Fα} be a set of homogeneous polynomials defining X. Parametrize a k-plane Λ via a linear map k n P → Λ ⊂ P , and consider the expressions Fα(Λ)—these are homogeneous polynomials in the parameters of Λ. Then Λ ⊂ X if and only if each Fα(Λ) vanishes identically with respect to the parameters. But the coefficients of the polynomials Fα(Λ) are themselves k n polynomials Cαβ on the space of linear maps P → P . Taking slightly greater care with these parametrizations (Remark 1.2 below), we can use elimination theory (see [18] Ch. 14, 15) to determine computationally whether or not the equations Cαβ = 0 have a simultaneous solution that corresponds to a k-plane Λ ⊂ X. Our initial question is thus answered for the given variety X, provided we can successfully perform these computations over our base field K. Such computational methods are not, however, the subject of this paper. Rather, we hope to determine general circumstances under which we can expect to find k-planes on varieties. And the previous argument suggests the point of entry for the full techniques of algebraic geometry: the set of k-planes contained in a given X form a projective variety, embeddable in the Grassmannian G(k, n), which we shall call the Fano Variety Fk(X) ⊂ G(k, n). This observation allows us to rephrase the initial question quantitatively: what is the dimension of the variety of k-planes Fk(X) lying on X? If we can establish that dim(Fk(X)) ≥ 0, then we can answer the original question to the affirmative. In this paper, we will restrict our attention to the natural first case, that of hypersurfaces n —this will allow us, crucially, to consider all hypersurfaces of degree d in P simultaneously N via their parameter space P . With this advantage, we will be able to give several answers that in fact depend on no more than this degree d as compared to the dimensions k and n, to which we will eventually add the requirement of smoothness. 0.1. Dimension of Fano varieties. Our first and most general claim concerning k-planes on hypersurfaces is as follows. Define k + d φ(n, d, k) = (k + 1)(n − k) − . d n Let X ⊂ P be a hypersurface of degree d ≥ 3. We claim that

(1) dim(Fk(X)) ≥ φ(n, d, k) if φ ≥ 0. For a general X, we claim that (1) is an equality if φ ≥ 0, and that

Fk(X) = ∅ if φ < 0.

If this holds for a particular X, we will say that Fk(X) has “the expected dimension.” Restated, our claim is that for a general hypersurface X, k + d codim (F (X) ⊂ (k, n)) = = #{degree d monomials on k}. k G d P

In the crude argument given above, this is the number of coefficients “Cβ” of “F (Λ),” the restriction to Λ of a defining polynomial F of the hypersurface X. So, this would appear i ii AUTHOR: ALEX WALDRON ADVISOR: JOE HARRIS to be the number of conditions cutting out Fk(X). However Fk(X) is a subvariety of the Grassmannian G(k, n), and therefore such an argument fails to show that Fk(X) is in fact non-empty if φ ≥ 0. Indeed, the requirement d ≥ 3 is essential, as shown in Remark 1.8. And, although the argument will be entirely classical—a dimension count via incidence correspondences (two, in this case)—the claim was established only through the result of Hochster and Laksov [1] in 1987. Interestingly, in the case φ(n, d, k) ≥ 0, the proof that Fk(X) is non-empty for all n X ⊂ P of a given degree d will depend on proving the existence of a hypersurface X0 such that Fk(X0) has dimension exactly φ(n, d, k) (see Remark 1.9). We will give several such examples, thereby proving particular cases of the claim. For arbitrary n, d, k, we will prove that a general surface has Fano variety of dimension φ; even while we are left with no way to exhibit such a hypersurface, nor with means to check that a particular hypersurface is “general” in our sense. This is the power of using “incidence correspondences” (section 1.2.1) in conjunction with the Theorem on Fiber Dimension (Proposition 1.3), in contrast to the naive computational argument suggested at the outset. 0.2. The low-degree limit. This last result establishes that all hypersurfaces have “enough” k-planes corresponding to their degree and dimension, and that a general hypersurface has the expected-dimensional family of k-planes. But the next question remains inscrutable: which hypersurfaces have “too many” k-planes, by which we mean a Fano variety of dimension greater than φ? We propose two criteria for remedying this situation, i. e. specifying which hypersurfaces are “general” in the previous sense. The first, smoothness, is natural especially when working over a field K of characteristic zero (as we will in Ch. 2 and 3). However, it is not sufficient. In the case k = 1 of lines, the canonical example of a smooth hypersurface fails: if char(K) = 0, the Fano variety of the Fermat n hypersurface of degree d = n + 1 in P has dimension n − 3 (see [5]), greater than the estimated dimension φ(n, d, 1) = 2n − 3 − d = n − 4. Second, we work in the limit of low degree compared to dimension. This is a realm of broad interest pertaining to several different fields: see for example Kollar’s article [12]. By itself, however, low degree is clearly insufficient for Fk(X) to have the expected dimension, as seen by considering any reducible hypersurface. Working in the low-degree limit is a common approach for “specializing” a known fact about general hypersurfaces to smooth ones.2 In the case of lines, k = 1, the well-known Debarre-De Jong conjecture [3] asserts that if d ≤ n and X is smooth, then dim(Fk(X)) = φ(n, d, 1) = 2n − 3 − d, i. e. X does not have “too many” lines. By the example just given, the bound d ≤ n is sharp, if it holds. This conjecture is actively pursued (see [3], [4]): the result has been established for d ≤ 6 [5], and for several months in 2007 a proof of the general case was thought to have been found. There has also been progress in the area of rational curves lying on hypersurfaces of low degree: Starr and Harris [10] show that for d < (n + 1)/2, a general degree d hypersurface contains the expected-dimensional variety of rational curves of each degree. n Our assertion is that for a fixed k, if X ⊂ P is smooth of degree d n, then X does not have too many k-planes. This is the result of Harris, Mazur and Pandharipande [2] in 1998, in which these same Fano varieties are also shown to be irreducible. The proof

2The term “specialization” can refer to a much more involved set of techniques having this function. FANO VARIETIES AND UNIRATIONALITY iii will be by induction, using a more delicate incidence correspondence and the technique of residual intersections.

0.3. Application to unirationality in low degree. Finally, we will turn to a question not obviously related to the previous two, that of unirationality. A variety X is said to be N unirational if there exists a dominant rational map P → X for some N—or, equivalently, if the function field K(X) is embeddable in a purely transcendental extension of the base field K. This property is a weakening of the important classical idea of rationality; but the latter notion has turned out to behave quite irrationally, even if we take char(K) = 0. All irreducible quadrics are rational ([13] 7.14). Elliptic curves are not, but smooth cubic 3 hypersurfaces in P are always rational. There are examples of smooth, cubic, rational projective hypersurfaces in all even dimensions, and there are known to exist the same in odd dimensions; but the general behavior in degree 3 is not known. Moreover, there are no smooth projective hypersurfaces of any degree d ≥ 4 that are known to be rational, nor has the possibility been ruled out (for much of this, see [13]). It was also long unknown whether unirationality is really a weaker notion than that of rationality for char(K) = 0: a curve is unirational if and only if rational, as is a surface, as was shown by Castelnuovo and Enriques. (Whereas if char(K) = p, the Zariski surfaces are unirational but not rational [8].) In 1972, however, Clemens and Griffiths showed 4 that most cubic threefolds in P are not rational—we will prove in Proposition 2.3 that smooth ones are unirational, and thus that unirationality is a strictly weaker notion in characteristic zero (see [13] Ch. 7). In contrast to the state of affairs concerning rationality, we will be able to show that n a smooth hypersurface in P of degree d n is unirational. This theorem has its roots in the same assertion regarding a general hypersurface, dating back to Morin in 1940 [7]. In 1992, Paranjape and Srinivas [6] clarified Morin’s proof and showed further that the general complete intersection of low multi-degree is unirational. Our result is again that of [2]: just as in the above result about Fano varieties, the unirationality result specializes to smooth hypersurfaces the earlier result for general ones. The construction of a rational “comb” of nested Grassmann bundles parametrizing the variety unirationally is similar to that of the earlier work, but in fact the crucial result above concerning Fano varieties allows the proof to hinge around smoothness.

0.4. Further work. We have yet to specify how we shall define “low degree.” The following table shows us the meaning of n d. (See sections 2.5 and 3.2.) These numbers n serve as follows: let X be a smooth hypersurface of degree d in P . If n ≥ N0(d, k) then X does not have “too many” k-planes, and if n ≥ U(d) then X is unirational. Judging by these high values,3 the results of this paper (from [2]) might be deemed a bit quixotic. The bound U(d) goes roughly as a 2d-fold iterated exponential of d! Admittedly, the greatest care has not been taken to find the minimum such bounds attainable by our inductive methods. Still, these are the first and remain the best known bounds of their kind for arbitrary d and k. However, following Harris et. al. [2], Jason Starr [10] has recently reduced two very closely related bounds to a combinatorial expression in d and k.

3A typo in the original paper [2] was discovered subsequent to these computations—these values should actually be slightly higher. iv AUTHOR: ALEX WALDRON ADVISOR: JOE HARRIS

Table 1. The meaning of n d.

N (d, k) d 0 U(d) k = 1 k = 2 k = 3 ··· 2 4 6 9 0 ··· 3 52 117 250 3 4 34276 366272 3391294 179124155 5 1017 1021 1025 10145 ··· 6 1082 10103 10122 108790 ......

k+d−1 Set M(d, k) := d−1 + k − 1. (This number will reappear in Lemma 2.11, for the reason that if n ≥ M(d, k) then X is swept out by k-planes.) In [9], it is shown using much more sophisticated techniques that if n ≥ M(d, k) + 1 and X is smooth, then Fk(X) has at least a component of the expected dimension. It is also shown there that if n ≥ M(d, k) then k the general k-plane section of X is a general hypersurface in P , whereas in [2] this same fact was proved only for n > N0(d − 1, k). So [9] has managed to improve one of the main results of our source [2] to a practical level, leaving one optimistic about such prospects for the results which we demonstrate here. Chen [11] has also shown as a corollary to [2] that the Fano variety is itself unirational in the low-degree limit. 0.5. Incidence correspondences. Perhaps the main idea of this paper is to take an “incidence correspondence” (see section 1.2.1) between two families in order to gain infor- mation about one by means of the other (often using the Theorem on Fiber Dimension (1.3)). An apology is even necessary for the repeated use of this idea in all three chapters— even the unirationally parametrizing variety will be built using the “relative Fano variety,” which is locally just a form of the incidence correspondence “I” of Chapter 1! Moreover, an idea similar to that of the first chapter is applied in the inductive step of our proof of unirationality: in Proposition 3.12 of Chapter 3, we will prove surjectivity by showing that the generic fiber of a map has appropriately low dimension, based on the fact from Ch. 2 that smooth hypersurfaces of low degree do not contain “too many” k-planes. (Although in Ch. 1 this argument is made for the projection onto the opposite factor.) The ubiquity of this construction becomes impressive. 1. Planes on Hypersurfaces in General. We show that a general hypersurface X of degree d ≥ 3 has Fano variety of dimension dim(Fk(X)) = φ(n, d, k), the number defined above. Remarkably, however, we will be left neither with a way to determine whether a given hypersurface has the expected- dimensional Fano variety, nor with a sure-fire means of producing examples in any particular case—even through in the complex case we could randomly choose such a hypersurface with probability 1 (if such a thing were possible). 1.1. Definition of the Fano variety. Let V be an (n+1)-dimensional vector space over ∼ n an algebraically closed field K, and let X ⊂ P(V ) = P be a projective variety. Define the k’th Fano Variety Fk(X) as the set of projective k-planes contained in X,

Fk(X) = {Λ ∈ G(k, n) | Λ ⊂ X}.

In this section we will show that Fk(X) is in fact a projective subvariety of the Grassman- nian, once we have recalled some basic facts. Vk+1 We will view G(k, n) as a closed subvariety of P( V ) via the Plucker embedding

Λ = hv0, . . . , vki 7→ [v0 ∧ · · · ∧ vk] . We will also occasionally refer to the Grassmannian G(k + 1, n + 1) of (k + 1)-dimensional linear subspaces of V , and of course G(k + 1, n + 1) = G(k, n). We will sometimes for emphasis write G(k + 1,V ) speciﬁcally for the Grassmannian of (k + 1)-planes in V, and likewise G(k, PV ) = G(k + 1,V ). (k+1)(n−k) Proposition 1.1. The variety G(k, n) is covered by open sets isomorphic to A . Therefore, dim(G(k, n)) = (k + 1)(n − k).

Proof. Decompose V as a direct sum V = V0 ⊕ W0, with dim(V0) = k + 1 and dim(W0) = n − k. Choose a basis {v0, . . . , vk, wk+1, . . . , wn} for this direct sum, and take coordinates on V dual to this basis. Let U ⊂ G(k + 1, n + 1) be the set of (k + 1)-dimensional linear subspaces of V complimentary to W0 (which includes V0). Then U is the intersection with Vk+1 G(k + 1, n + 1) of the affine patch of the ambient projective space P( V ) consisting of points whose (v0 ∧ v1 ∧ · · · ∧ vk)-coordinate is nonzero. Hence U is open and affine. For a (k + 1)-plane Λ ∈ U, define the vectors Λi = (vi + W0) ∩ Λ, for i = 0 . . . k. Then {Λi} is a basis for Λ, and we can represent Λ by the (k + 1) × (n − k) matrix MΛ with rows Λi:   1 0 ··· 0 a11 ··· a1(n−k)  0 1 ··· 0 a21 ··· a2(n−k)  (2) M =   . Λ  . . . . .   . . . . .  0 0 ··· 1 a(k+1)1 ··· a(k+1)(n−k) The Plucker coordinates on U are the (k + 1) × (k + 1) minors of this matrix. One can see that each entry aij of the (k + 1) × (n − k) submatrix (aij) is in fact the value of some ∼ (k+1)(n−k) one of these minors. So, the aij are in fact affine coordinates on U = A . Each affine coordinate patch (i. e. complement of a coordinate hyperplane) of the ambient space meets G(k, n) in a set U of this form, which implies that dim(G(k, n)) = (k+1)(n−k). 1 2 AUTHOR: ALEX WALDRON ADVISOR: JOE HARRIS

Thus in order to show that Fk(X) is a closed subvariety of G(k, n), it is sufficient to show that Fk(X) ∩ U is an affine variety, for any such affine patch U ⊂ G(k, n). Wlog, let X be a hypersurface defined by a single homogeneous polynomial f(X0,...,Xn) of degree d; the general case will follow just by taking intersections. For X a hypersurface, Λ ⊂ X iff the restriction of f to Λ vanishes identically. Let Λ ∈ U, and parametrize the linear P space Λ by [u0, . . . , uk] 7→ [ uiΛi] , with Λi the i’th row of the matrix MΛ from (2) (i. e. basis vector for Λ). P We then simply plug in this parametrization to obtain a polynomial f( uiΛi) of the same homogeneous degree d in the parameters {ui} of the space Λ. The coefficients are then inhomogeneous polynomials in the aij; their vanishing is necessary and sufficient for f to vanish identically on Λ, i. e. for the space Λ to be contained in the locus {f = 0} = X. Thus we have obtained polynomial equations in the aij defining {Λ ∈ U | Λ ⊂ X} = Fk(X) ∩ U, for any such U. Hence Fk(X) is a closed subvariety of G(k, n), and a projective variety itself. Note that if X is quasi-projective, with projective closure X, then the Fano Variety Fk(X) ⊂ Fk(X) is a locally closed subvariety. This is simply because for any two families of subvarieties of a variety, the subset of disjoint pairs is open ([13], Ch. 4). Remark 1.2. Elimination theory can be used on each element of a finite cover of patches ∼ (k+1)(n−k) {Ui = A }, using the explicit polynomial description of Fk(X) ∩ Ui just given, to answer the opening question of whether or not a given variety X actually contains a k-plane.

1.2. Estimating the dimension of Fk(X).

1.2.1. Incidence correspondence. In order to find an estimate of the dimension of Fk(X) for a hypersurface, we set up an incidence correspondence between hypersurfaces and k- N planes. Let P be the projective space of homogeneous polynomials of degree d in n + 1 n+d variables, so N = d − 1. Then let N I = {(f, Λ) ∈ P × G(k, n) | f(Λ) = 0}. n Given any two families of projective varieties in P , the set of pairs such that the first subvariety is contained in the second is a constructible set (see [13]). Here, I is in fact N a closed subvariety of P × G(k, n), which can be shown along much the same lines as N the above argument for Fk(X). Choose an affine coordinate patch U0 of P × G(k, n). N Under the Segre embedding, U0 = U1 × U2 for two affine coordinate patches U1 ⊂ P and U2 ⊂ G(k, n). Then U1 is of the form

N h0 X h U1 = {[f(T )] ∈ P | f(T ) = T + ehT }, h6=h0 and we let U2 ⊂ G(k, n) have affine coordinates aij coming from the matrix MΛ of (2). Then via the Segre embedding, one obtains inhomogeneous coordinates on U0 = U1 × U2 as pairs ({eh}, {aij}). Meanwhile, the points f ∈ U1 are still homogeneous polynomials, and the points Λ ∈ U2 are still projective linear spaces. So we can parametrize Λ as before by [u0, . . . , un] 7→ P [ u`Λ`]. Then f(Λ) is a homogeneous polynomial in the parameters u`, whose coefficients are doubly inhomogeneous polynomials in the coefficients eh of f and the entries aij of the FANO VARIETIES AND UNIRATIONALITY 3 matrix MΛ. These coefficients vanish exactly when f(Λ) ≡ 0, so they define I as a variety N on the affine coordinate patch U0. Hence I is a projective subvariety of P × G(k, n), as desired.

1.2.2. Dimension estimate. Now we can use I to estimate the dimension of Fk(X) for a hypersurface X. Consider the restriction to I of the projections onto the factors of N P × G(k, n), I ~ GG π1 ~ GG π2 ~~ GG ~~ GG ~ ~ G# # N P G(k, n). N The fiber of π1 over a point X ∈ P is the Fano variety Fk(X). So we hope to estimate this dimension using the following fundamental result. Recall that a function f : X → R −1 is upper-semicontinuous if f ([r, ∞)) is closed for all r ∈ R. If furthermore f : X → Z takes integer values, then f attains its minimum on an open set of X. Proposition 1.3. (Theorem on fiber dimension) Let X be quasi-projective and ϕ : X → n P a regular map, and let Y = ϕ(X) be the closure of the image. For each p ∈ X, let −1 Xp = ϕ (ϕ(p)) be the fiber of ϕ through p. Then, (a) For all p ∈ X, dimp(Xp) ≥ dimp(X) − dimϕ(p)(Y ), with equality on a nonempty open subset of X. The function dimp(Xp) is upper-semicontinuous on X, i. e. for any integer k the set of p such that dimp(Xp) ≥ k is closed. (b) If X is projective, then for all q ∈ Y, dim(ϕ−1(q)) ≥ dim(X) − dim(Y ), with equality on a nonempty open subset of Y. The function dim(ϕ−1(q)) is an upper- semicontinuous function of q ∈ Y. Proof. This is Theorem 11.12 and Cor. 11.13 of [13], and Section I.6.3, Theorem 7 of [15].

Corollary 1.4. (a) Let X0 be an irreducible component of X,Y0 = ϕ(X0), and µ = minp∈X0 (dimp(Xp)). Then dim(X0) = dim(Y0) + µ. −1 (b) For X projective, let X0 be an irreducible component and let λ = minq∈Y (dim(ϕ (q))). Then dim(X0) = dim(Y0) + λ. Proof. These are attached to the theorem in [13] and [15]. Corollary 1.5. Let X be an irreducible quasi-projective variety, and ϕ : X → Y be a dominant morphism. If any ﬁber of ϕ contains an isolated point, then dim(X) = dim(Y ) and the general ﬁber of ϕ has dimension zero.

Proof. Let p ∈ Xp be an isolated point. Then

dimp(Xp) = 0 = dim(X) − dim(ϕ(X)) = dim(X) − dim(Y ), by Corollary 1.4(a). The second claim is immediate from 1.3(a). 4 AUTHOR: ALEX WALDRON ADVISOR: JOE HARRIS

The results of this chapter will come from applying these statements to the map π1 as well as in several similar instances. First, we apply Proposition 1.3 to the map π2 to compute the dimension of I. Given Λ ∈ G(k, n) consider the surjective linear map n ∼ k {polynomials of degree d on P } −→ {polynomials of degree d on Λ = P } n+d given by restriction to Λ. The kernel of this map is a linear subspace of dimension d − k+d d , whose projectivization is the the ﬁber of π2 over Λ. Therefore the ﬁbers of π2 are all irreducible of the same dimension, so I is irreducible (by Theorem 11.14 of [13]). From Proposition 1.3, n + d k + d dim(I) = dim( (k, n)) + dim(π−1(Λ)) = (k + 1)(n − k) + − − 1. G 2 d d

The map π1 cannot be dealt with so straightforwardly. However, it will be shown in section 1.4 that its image has maximal dimension (unless d = 2). This means that π1 is injective when dim(I) ≤ N, and π1 is surjective when dim(I) ≥ N. Once we establish this, we will be able to count the dimension by applying Proposition 1.3. Until this last fact was proven, by [1] in 1987, the following was merely a “dimension estimate:” Theorem 1.6. For n ≥ k ≥ 0 and d ≥ 3, let k + d φ(n, d, k) = dim(I) − dim( N ) = (k + 1)(n − k) − . P d N (a) For φ(n, d, k) < 0, the subvariety of P of hypersurfaces that contain a k-plane has codimension −φ. n (b) For φ(n, d, k) = 0, every hypersurface of degree d in P contains a k-plane, and a general hypersurface contains a positive number of k-planes. (c) For φ(n, d, k) > 0, a hypersurface X has dim(Fk(X)) ≥ φ(n, d, k), with equality for general X. By the preceding discussion, this theorem will be proved if we can establish that the N image of π1 : I → P has maximal dimension, i. e. is surjective for φ ≥ 0 and injective for φ ≤ 0. At present, however, we can prove only small pieces of this theorem. It is clear that for φ < 0, the variety of hypersurfaces that do contain a k-plane has codimension at N least −φ = dim(P ) − dim(I), so the following is already evident: Corollary 1.7. For φ < 0, a general hypersurface contains no k-planes. Also note that for φ > 0, if a hypersurface X contains a k-plane (i. e. lies in the image −1 π1(I)) then it contains inﬁnitely many k-planes, since dim(Fk(X)) = dim(π1 (X)) ≥ φ(n, d, k) > 0. The nontrivial task is to show that a hypersurface does in fact contain a k-plane: 4 Example 1.8. The general quadric threefold in P contains no 2-planes—this is the case n = 4, k = 2, d = 2 and φ(4, 2, 2) = 0. For, in the family of all quadric hypersurfaces, the sub-family of cones over quadric surfaces has codimension 1; and each such cone contains at least a 1-dimensional family of 2-planes. Therefore this sub-family must equal the image π1(I). The stipulation d ≥ 3 is thus necessary in (b), and as such it is not obvious that (b) holds at all. FANO VARIETIES AND UNIRATIONALITY 5

Meanwhile, in case (c) there is a trivial partial solution to this problem: for n ≥ d+k−1 n d−1 + k − 1, any hypersurface X ⊂ P is swept out by k-planes (by Lemma 2.11 to follow). This prevents such a failure as occurs for quadric threefolds in Example 1.8: the N N map π1 : I → P is surjective, therefore dim(Fk(X)) ≥ dim(I) − dim(P ) = φ(n, d, k) with equality for general X. This suggests the fruitfulness of the low-degree limit, to which we will turn in subsequent chapters. But below this point, the existence of even a single k-plane is nontrivial. The method of proof will be as follows. In order to establish parts (a) and (b) of Theorem 1.6, we will use Corollary 1.5: we must simply demonstrate the existence of a hypersurface whose Fano scheme has an isolated point. The fact that such hypersurfaces exist in general will be shown fully in Section 1.4. And similarly, Proposition 1.3 will imply (c) if we can prove the existence of a hypersurface X containing a k-plane Λ such that dimΛ(Fk(X)) = φ, i. e. that Fk(X) has the expected dimension locally at Λ. In section 1.4, we will show that the existence of such an X is implied by the full algebraic result of [1]. In summary, Remark 1.9. To establish Theorem 1.6 for φ ≤ 0 (parts (a) and (b)), it is sufficient to show the existence of a hypersurface X0 whose Fano variety contains an isolated point. To establish the theorem for φ ≥ 0 (part (c)), we need dim(Fk(X0)) = φ. 1.3. Proofs by example. Here we give some “proofs by example” of Theorem 1.3(b) for particular choices of n, d, k for which φ(n, d, k) = 0, i. e. we exhibit hypersurfaces whose Fano variety has an isolated point. By Corollary 1.5 to the Theorem on Fiber Dimension, n these are, in fact, proofs that all degree d hypersurfaces in P contain k-planes. 1.3.1. The case k = 1. We first attend to the case of lines. Example 1.10. Let k = 1, and let n and d be integers such that φ(n, d, 1) = 2n − 3 − d = 0. n One can readily exhibit a degree d hypersurface in P with an isolated line. Note that (d− n 1)/2 = n−2, and take coordinates [Z0,Z1,W0,...Wn−2] on P . Consider the hypersurface X0 defined by the polynomial P(d−1)/2 d−1−k k F (Z,W ) = k=0 WkZ0 Z1 d−1 2 d−1−2 d−1 = W0Z0 + W0Z0 Z1 + ··· + Wn−2Z1 .

Then `0 = {Wi = 0} is an isolated line of Fk(X0). For, taking coordinates aij (i = 0, 1 and j = 1, . . . n − 1) on the aﬃne space U of lines ` complementary to the (n − 2)-plane {Z0 = Z1 = 0}, we get d−1 d−1−2 2 d−1 F (`) = (a01Z0+a11Z1)Z0 +(a02Z0+a12Z1)Z0 Z1 +···+(a0(n−1)Z0+a1(n−1)Z1)Z1 .

Then clearly `0 = (aij = 0) is the only line in U such that F (`0) = 0, hence `0 is an isolated point of the Fano variety. As one case, we have proved the following: 3 Corollary 1.11. A cubic hypersurface in P contains lines. Moreover, it is possible to obtain parts (a) and (c) of Theorem 1.6 in the case k = 1 by appropriately truncating or adding terms to the polynomial F (Z,W ). 6 AUTHOR: ALEX WALDRON ADVISOR: JOE HARRIS

1.3.2. The case k = 2 and above. We give similar examples in the cases k = 2, d = 4, 5 and φ = 0. From these examples we can inductively describe polynomials as required for all d ≡ 1, 2 mod 3, proving all cases k = 2 and φ = 0. Example 1.12. 4 (a) (n = 7, d = 4, k = 2) The quartic hypersurface in 7 P = {[Z0,Z1,Z2,W0,W1,W2,W3,W4]} defined by the equation 3 3 3 2 2 2 W0Z0 + W1Z1 + W2Z2 + W3Z0Z1Z2 + W4(Z0Z1 + Z1Z2 + Z2Z0 ) = 0, contains the 2-plane {Wi = 0} as an isolated point of its Fano variety of 2-planes. 9 (b) (n = 9, d = 5, k = 2) The quintic hypersurface in P = {[Z0,Z1,Z2,W0,...,W6]} defined by the equation X i j k 2 2 2 W`Z0Z1Z2 + W6(Z0 Z1Z2 + Z0Z1 Z2 + Z0Z1Z2 ) = 0 i,j,k even contains the 2-plane {Wi = 0} as an isolated point of its Fano variety of 2-planes. 8 (c) (n = 8, d = 3, k = 3) The cubic hypersurface in P = {[Z0,...,Z3,W0,...,W4]} defined by the equation 3 X 2 WiZi + W4(Z0Z1 + Z1Z2 + Z2Z3 + Z3Z0) = 0 i=0 contains the 3-plane {Wi = 0} as an isolated point of its Fano variety of 3-planes. P To prove these assertions, one must show that upon substituting Wi = aijZj the resulting forms span K [Z]d as aij varies. Then, since φ(n, d, k) = 0, they are also linearly independent, meaning that Λ = (aij = 0 ∀ i, j) is the only k-plane contained in the affine patch U ∩ Fk(X). This is seen in part (a) by choosing values of the coefficients aij as follows: setting Wi = Zi for each i = 0, 1, 2 individually, we see that all of the monomials 4 Zi are included in the span. Setting W3 = Zi for each i individually, we get each monomial 3 involving all of the Zk. Now, setting W4 = Zi, the only monomial not divisible by any Zk 2 2 or involving all three Zi is Zi Zi+1, and these are exactly the remaining monomials. So we have shown that all the monomials of K [Z]4 are in the span. Parts (b) and (c) are treated similarly. For the case k = 2, one can describe inductively a set of polynomials defining hypersurfaces with the required property. We have treated the cases d = 4, 5 individually in Example 1.12(a), (b), respectively. Then, given Fd(Z,W ) of degree d as required, define 0 0 d+2 0 d+2 0 d+2 Fd+3(Z, W, W ) = Z0Z1Z2Fd(Z,W ) + WaZ0 + WbZ1 + WcZ2 d X 0 k+1 d+1−k k+1 d+1−k k+1 d+1−k + Wk(Z0 Z1 + Z1 Z2 + Z2 Z0 ). k=1

4 P Part (a) here was found by writing out the the result of substituting Wi = aij Zj into an arbitrary polynomial, and choosing its coeﬃcients so that the result spans K [Z]4 . Part (b) was found by writing out the degree d monomials in a 2-simplex with the degree d − 1 monomials interlaced, and choosing a polynomial that spans K [Z]5 under certain choices of coeﬃcients aij . Part (c) was found similarly. FANO VARIETIES AND UNIRATIONALITY 7

0 This will again deﬁne a hypersurface having {Wi = Wi0 = 0} as an isolated 2-plane. This proves Theorem 1.6(b) in the case of 2-planes. In the case d = 3, k = 2, it is possible to ﬁnd a polynomial whose Fano variety has local dimension 2. Then the same formula can be used to extend to all d ≡ 0 mod 3. It is not clear how to generalize either the base cases or the induction method to k > 2.

1.4. Fk(X) has the estimated dimension for general X. In this section we prove Theorem 1.6 (a) and (b) in general, and show that (c) is implied by the main theorem of [1]. 1.4.1. An algebraic lemma. The following lemma, a partial version of the result of [1], will show directly the existence of the polynomials we require. We adopt notation similar to that of [1]. Let K [x] be the ring of polynomials in r variables x1, . . . , xr. Let d + r − 1 N(r, d) = dim (K [x] ) = K d d be the dimension of the homogeneous part of degree d (unrelated to the number N(d, k) defined later in Section 2.5). In the proof of the lemma we will use several other spaces from [1]. Let P be the m projective space of m × r matrices with entries in K. Let B = (K [x]d−1) be the affine t space of m-tuples of degree d − 1 forms in r variables. Let ~x = (x1, . . . , xr) be the column vector with the formal variables as entries. If [A] ∈ P is represented by the matrix A, then bA~x is a 1 × 1 matrix whose unique entry is a polynomial expression of homogeneous degree d in the variables xi. In the lemma, we will use a strategy surprisingly similar to the central proof of the chapter. Let V = {(b, [A]) ∈ B × P | bA~x = 0}, a closed subvariety of B ×P. Then a form b = (F1,...Fm) is in the image of the projection π1(V ) onto the first factor if and only if there exists a nontrivial linear dependence between m×r the mr forms {xjFi}, i. e. bA~x = 0 for some nonzero matrix A ∈ K . We will use V to bound the dimension of the subvariety of B consisting of m-tuples that result in such a linear dependence. Finally, let Pt ⊂ P be the irreducible quasi-projective subvariety of P consisting of points representable by a matrix of rank exactly t (see [13]), and let Vt = V ∩ (B × Pt). Lemma 1.13. Let d ≥ 3 and m ≥ r be positive integers such that N(r, d) = mr + ` for some nonnegative integer `. Let Fi, i = 1, . . . m, be generic forms in K [x]d−1 . Then the mr forms {xjFi}, for i = 1, . . . , m and j = 1, . . . r, are linearly independent in K [x]d . Proof. This claim comes from the main idea of [1] by a much shorter argument, due to the restriction ` ≥ 0 (which amounts to φ ≤ 0). Let π2 be the restriction of the projection onto the second factor V,→ B × P → P. It certainly may be possible that V is reducible, which makes the situation here nontrivial. Let 0 ≤ t ≤ r, so π2 restricts to a map Vt → Pt. We will apply the Theorem on Fiber Dimension to these maps to eventually determine that (3) dim(V ) = mN(r, d − 1) − 1, 8 AUTHOR: ALEX WALDRON ADVISOR: JOE HARRIS i. e. dim(V ) = dim(B) − 1, and a generic m-tuple (Fi) ⊂ B will satisfy the conclusion of the lemma. −1 Let p = [(aij)] ∈ Pt be represented by the rank t matrix (aij), and let F = π2 (p) be the fiber of π2 over p. We will first show that this fiber is irreducible and has dimension (4) dim(F ) = mN(r, d − 1) − N(r, d) + N(r − t, d). Since its unique P -coordinate is p, the variety F may be identified with its projection m onto B, i. e. the set of vectors b = (F1,...,Fm) ∈ (K [x]d−1) such that b(aij)~x = 0. So t letting L be the column vector of linear forms (aij)~x = L = (L1,...,Lm) , we think of F as the affine cone {b ∈ B | b · L = 0}. So F is simply a linear subspace of the vector space B, hence irreducible. Let J ⊂ k [x] be the ideal generated in L1,...,Lm. Then we have the short exact sequence of graded rings

(K [x](−1))m →·L K [x] → K [x] /J. Since F = Ker(·L :(K [x](−1))m → K [x]), this induces the short exact sequence of vector spaces m L 0 → F → (K [x]d−1) → K [x]d → (K [x] /J)d → 0. Since K [x] /J is isomorphic to a polynomial ring in r − t variables, this sequence directly yields the desired dimension count for dim(F ) in (4). Now, notice that the map π2 is surjective onto P , since (0, p) ∈ V ∀ p ∈ P. Furthermore, dim(Pt) = mr − 1 − (m − t)(r − t) = N(r, d) − ` − 1 − (m − t)(r − t) by [13] Prop. 12.2 and the given. So, since the ﬁber F was arbitrary, from Corollary 1.4 to the Theorem on Fiber Dimension we have

dim(Vt) = dim(F ) + dim(Pt) = mN(r, d − 1) − N(r, d) + N(r − t, d) + N(r, d) − ` − 1 − (m − t)(r − t) = (mN(r, d − 1) − 1) + N(r − t, d) − (m − t)(r − t) − `. To show 3, we now must prove that (m − t)(r − t) + ` ≥ N(r − t, d) for all t ≤ r, with our choice of m and r. Proceed by induction on t: the base case t = 0 follows from the assumption N(r, d) = mr +`. Now assume (m−(t−1))(r −(t−1))+` ≥ N(r −(t−1), d): r − t (m − t)(r − t) = (m − t)(r − t + d) r − t + d r − t ≥ (m − t + 1)(r − t + 1) since m ≥ r ≥ t and d − 1 ≥ 2 r − t + d r − t ≥ (N(r − t + 1, d) − `) by induction r − t + d ≥ N(r − t, d) − `.

We have established that for all t ≤ r, dim(Vt) ≤ dim(Vr) = mN(r, d − 1) − 1. Since S V = Vt, we get dim(V ) = maxt(dim(Vt)) = dim(Vr) = mN(r, d − 1) − 1, as desired. FANO VARIETIES AND UNIRATIONALITY 9

We have shown mN(r, d − 1) − 1 = dim(V ) < mN(r, d − 1) = dim(B), so π1(V ) must n be a proper subvariety of B = (K [x]d−1) . Thus an m-tuple (F1,...,Fn) ∈ B − π1(V ), i. e. a generic n-tuple, yields a linearly independent set {xjFi} in K [x]d .

1.4.2. Proof of Theorem 1.6. Recall that as per Remark 1.9, in order to prove Theorem 1.6 we must simply show the existence of certain hypersurfaces. Let d and n > k positive integers, and take coordinates [Z0,...,Zk,W1,...,Wn−k] on n P . We will restrict our attention to a hypersurface X0 deﬁned by a polynomial of the form n−k X F (Z,W ) = WiFi(Z0,...,Zk), i=1 where each Fi is a homogeneous polynomial of degree d − 1. The hypersurface X0 then contains the k-plane Λ0 = {W1 = W2 = ··· = Wn−k = 0}. Consider the aﬃne patch U of the Grassmannian consisting of k-planes complementary to the (n − k − 1)-plane {Z0 = ··· = Zk = 0}. Here we have coordinates aij from (2), with Λ0 corresponding to aij = 0. A plane Λ ∈ U is parametrized as h X X i [Z0,...,Zk] 7→ Z0,...,Zk, a1jZj,..., a(n−k)jZj .

The restriction F |Λ is then given simply by

n−k X X F |Λ(Z0,...,Zk) = ( aijZj)Fi(Z0,...,Zk), i=1 j and a hyperplane Λ lies on X0 if and only if this restriction vanishes identically. Now, assume that the Fi are generic forms in K [Z]d−1 , and φ(n, d, k) = (k + 1)(n − k+d k) − d ≤ 0. The conditions of Lemma 1.13 are met if we take m = n − k, r = k + 1, for then r + d − 1 k + d N(r, d) = = ≥ (k + 1)(n − k) = rm. d d

So from the lemma we may conclude that the (k + 1)(n − k) forms {ZjFi(Z)} are linearly independent in K [Z]. But this is exactly the statement

n−k X X F |Λ(Z) = ( aijZj)Fi(Z0,...,Zk) ≡ 0 iﬀ aij = 0 for all i, j. i=1 j

Thus Λ0 = (aij = 0 ∀ i, j) is the unique point of Fk(X) ∩ U, hence an isolated point, as desired. This establishes parts (a) and (b) of Theorem 1.6. The full result of [1] adds to Lemma 1.13 the fact that for N(r, d) > rm, the linear map

rm ∼ ∼ N(r,d) K = K [Z]1 ⊗ K [Z]d−1 → K [Z]d = K Zj ⊗ Fi(Z) 7→ ZjFi(Z) 10 AUTHOR: ALEX WALDRON ADVISOR: JOE HARRIS is surjective, for d ≥ 3.5 Hence the kernel of this map has dimension rm − N(r, d) = φ(n, d, k) in our set-up, which is exactly the statement that F |Λ(Z) ≡ 0 for aij in a linear ∼ rm subspace of dimension φ(n, d, k). Thus Fk(X0) ∩ U ⊂ U = A is a linear subspace of dimension φ, which is therefore the local dimension of Fk(X0) on U. This establishes part (c) of the theorem. Remark 1.14. It is somewhat surprising that our argument establishes surjectivity only in the case φ(n, d, k) = 0, and not for the case that the dimension of I is strictly bigger N than that of the target P . However, this is not the only failure of its kind, e. g. a surjective map of sheaves is not in general surjective on global sections unless it is also injective.

5The requirement d ≥ 3 is not included in [1]. The error is simply the unjustified claim that the inequality at the bottom of p. 237 holds for “d = 1” (which is d − 1 = 1 in this paper). The subsequent combinatorial proof of the inequality for d ≥ 2 is correct. 2. Fano Varieties in the Low-Degree Limit. The goal of this chapter will be to specify when a hypersurfaces is “general” in the sense of the previous chapter, meaning that its Fano variety of k-planes has the expected dimension φ (the number defined in Theorem 1.6). Our answer will be that provided n d n, this holds for all smooth hypersurfaces of degree d in P . Our main reference for this chapter and the next is the 1998 paper [2] by Harris, Mazur, and Pandharipande. 2.1. Notation and terminology. Throughout the rest of the paper, we work over a fixed algebraically-closed field K of characteristic 0. By a scheme, we will mean a quasi- n projective K-scheme, i. e. a locally closed subscheme of P . A reduced scheme will then be called a variety (not necessarily irreducible), which is therefore just a classical quasi- projective variety. So for our purposes, an integral K-scheme is an irreducible quasi- projective variety. Furthermore, we will have occasion to consider only closed (i. e. geometric) points of schemes: by a “point” x ∈ X we will mean a closed point. If W and T are B-schemes, we will abbreviate WT = W ×B T for their fiber product. Definition 2.1. A reduced B-scheme P → B will be called a projective bundle of rank r r, or a “P -bundle,” if for each b ∈ B there exists a Zariski-open neighborhood U 3 b yielding a commutative diagram

∼ r PU / U × H P HH HH HH HH$ U. Remark 2.2. (a) In this paper, all projective bundles will be sub-bundles of some trivial n bundle B × P = B × PV. For our purposes, the operation“P” will send a vector sub- bundle of B × V to the corresponding projective sub-bundle of B × PV (which is simply Grass0(E) as defined below). Thus every projective bundle P will be equal to P(E) for a vector sub-bundle E ⊂ B ×V. Where convenient, we will thus be able to work equivalently in the context of vector bundles. (b) There frequently arise bundles X → B such that each fiber Xb is isomorphic to a r projective space P , while the bundle fails to be locally trivial in the Zariski topology: an example is the family of smooth conics in the plane, which does not even admit a rational section (see [13], ch. 4). In fact, this failure can be construed as the main difficulty encountered in Chapter 3. However, if the given projective bundle is a sub-bundle of a trivial bundle, i. e. X ⊂ n r n B × P → B, and the scheme-theoretic fibers are reduced linear subspaces P ⊂ P , then X → B is in fact locally trivial: this is Proposition 3.3, to follow. 2.2. A first result. To give a first demonstration of the techniques of the remainder of the paper (residual varieties in particular) we will prove a well-known result. The following is actually the simplest case of the main result of the next chapter, Theorem 3.6; but the techniques are quite similar to those used in the proof of the main result of this chapter, Theorem 2.13—which will in turn be necessary for the proof of Theorem 3.6. The proof crucially involves the question of whether a bundle with projective fibers is actually a “projective bundle” as per Definition 2.1. 11 12 AUTHOR: ALEX WALDRON ADVISOR: JOE HARRIS

n Proposition 2.3. For n ≥ 3, a smooth cubic hypersurface X ⊂ PK is unirational, provided K is algebraically closed and char(K) = 0. Proof. Certain notions used here (such as Grassmann bundles) will be made rigorous in Section 2.3 immediately following this proof, which will thus be somewhat casual. From Theorem 1.6, we may choose a plane Γ of some dimension l > 0 lying on X (a line, if one prefers). Then the variety of (l + 1)-planes containing the l-plane Γ is parametrized n−l−1 n by P = P /Γ. Since X is smooth, a general such (l + 1)-plane Θ ⊃ Γ meets X in a the union of the plane Γ and a “residual” quadric hypersurface of dimension l, call it QΘ. n−l−1 So, X is birational to the total space of a “quadric bundle”—i. e. a P -scheme whose n−l−1 ﬁbers are quadrics—over an open subset of P .

To be precise, the quadrics QΘ are the generic fibers of the blow-up ˜ n−l−1 πΓ : X = BlΓ(X) −→ P , n−l−1 ∼ n−l−1 which we view as a quadric bundle over P via the projection πΓ from Γ = P . The proper transform BlΓ(Θ) of an (l + 1)-plane Θ ⊃ Γ is again an (l + 1)-plane, and QΘ = BlΓ(Θ) ∩ X.˜ (In the rest of this argument we will sometimes fail to mention when a property holds only over an open, dense subset.) We will show that X˜ is unirational by viewing it as a quadric bundle. Each irreducible quadric QΘ is rational individually: projecting from any point of QΘ to a hyperplane of Θ is 1-1 and dominant, therefore birational since char(K) = 0. We hope to piece together these rational parametrizations consistently, in order to parametrize an open subset of X.˜ The key to doing this is to find a rational section of the quadric bundle X˜ = {QΘ | Θ ∈ n−l−1 n−l−1 ˜ P }, i. e. a rational map σ : P 99K X with σ(Θ) ∈ QΘ. If we could find such a section σ, then we would have a rational map

(5) qΘ 7→ σ(Θ), qΘ ∈ G(1, Θ), deﬁned for each Θ. These maps are now clearly compatible over X˜ − E. Now, the spaces G(1, Θ) form the “Grassmann bundle of lines” (which we will soon refer to as “Grass1”) n−l−1 in the projective bundle of (l + 1)-planes Θ, parametrized by P . Any Grassmann bundle over a rational base is rational (see Proposition 3.5, to follow). This rational map from X˜ into the Grassmann bundle of G(1, Θ)’s is 1-1 and dominant over each Θ, hence is birational overall. In short, this paragraph has shown: FANO VARIETIES AND UNIRATIONALITY 13

Lemma 2.4. The total space of a family of pointed quadrics over a rational base is rational. However, there is no guarantee of finding such a section σ, i. e. a choice of point n−l−1 qΘ ∈ QΘ varying regularly with Θ ∈ P . Say we take l = 1 and look only for points of QΘ that lie inside the line Γ : then in general ZΘ := QΘ ∩ Γ will consist of two points, and we will have no way to pick one consistently. Thus we are forced to abandon hope of a rational parametrization, which is 1-1, and instead hope for a unirational parametrization, N i. e. a dominant map P 99K X—or, equivalently, a dominant map from a rational variety 3 to X. (Although in fact, smooth cubic surfaces in P are rational.) Now, by analogy we hope to find a “unirational section” of the bundle of the quadric bundle X˜ = {QΘ}, i. e. a map from a rational variety to X˜ that hits an open subset n−l−1 of Θ ∈ P . Fortunately, we will find such a rational variety by directly fixing the problem of the previous paragraph (using a familiar construction): consider the incidence correspondence n−l−1 Ψ = {(Θ, p) | p ∈ ZΘ = QΘ ∩ Γ} ⊂ P × Γ. This variety Ψ manifestly has a “unirational section” to X,˜ given by (Θ, p) 7→ p ∈ QΘ. And Ψ is itself a rational variety, which we will prove shortly in Lemma 2.5. What good is a “unirational section?” It furnishes a rational section not of X˜ but of a quadric bundle that dominates X,˜ namely the pullback

H := X˜ × n−l−1 Ψ. P This variety H is a quadric bundle over Ψ, whose fiber over (Θ, p) is by definition the pointed quadric (p ∈ QΘ). Thus by Lemma 2.4, H is rational if Ψ is rational. We can describe H more concretely as follows: ˜ n−l−1 H = {(q, Θ, p) | q ∈ QΘ, p ∈ ZΘ} ⊂ X × P × Γ. So H clearly dominates X˜ and, pending the following result, we have obtained a unirational parametrization of X˜ and hence of X. Lemma 2.5. The incidence correspondence Ψ is rational. n−l−1 Proof. The variety Ψ ⊂ P ×Γ is (over an open subset) a bundle of (l−1)-dimensional n−l−1 quadric hypersurfaces ZΘ = YΘ ∩ Γ ⊂ Γ over P , so it seems at first that we have made no progress towards a proof. However, consider Ψ as as a bundle over its other n−l−1 factor, Γ : observe that a point p ∈ Γ simply imposes a linear condition on Θ ∈ P , which corresponds to the tangent hyperplane X to X at p. In fact, Ψ = {Z } n−l−1 Tp Θ Θ∈P ∼ l n−l−1 is in fact a linear system of hypersurfaces in Γ = P , parametrized by Θ ∈ P . (This will be shown explicitly in Section 2.4.) Furthermore, this system has no basepoints, since a basepoint would be a singular point of X (see also 2.4). Therefore each fiber of the n−l−1 n−l−2 ∼ l projection Ψ → Γ is a hyperplane in P , so Ψ is a P -bundle over Γ = P (by Proposition 3.3 to follow), hence rational. 2.3. Constructions. We proceed to introduce several objects of which we will make extended use in this chapter and the next. For the duration of Section 2.3, we let E → B be a vector bundle of rank r + 1, with B and E irreducible varieties (i. e. integral quasi-projective K-schemes, in our terminology). 14 AUTHOR: ALEX WALDRON ADVISOR: JOE HARRIS

2.3.1. The relative Grassmannian. It is possible to define the relative Grassmannian variety Grassk(E) → B, whose fiber over each closed point b ∈ B is the ordinary Grassmannian G(k + 1,Eb)—one can simply define Grassk(E)U = U × G(k + 1, r + 1) over each open subset U ⊂ B such that π−1(U) =∼ U × Kr+1, pending a compatibility check. As mentioned already, we shall restrict our attention to the case that E is a sub-bundle of a trivial bundle B × V, with V an (n + 1)-dimensional vector space. Then we may simply define Grassk(E) = {(b, Λb) | Λb ⊂ Eb} ⊂ B × G(k + 1,V ). On each open subset B × Uα, with Uα an affine patch as in Proposition 1.1, Grassk(E) is defined as a subvariety by the requirement that each row of the matrix MΛ lie within the linear space Eb. 0 Given a subbundle E0 ⊂ E of rank r ≤ r, we will also consider the subvariety Grassk(E; E0) ⊂ Grassk(E) of (k + 1)-planes containing E0, which is isomorphic to Grassk−r0 (E/E0). If P = P(E) is the projective bundle corresponding to the vector bundle E, then we will write Grassk(P ) for the variety of projective k-planes over B, which is identical to Grassk(E). We will also let Grassk(P ; P0) = Grassk(E; E0) be the variety of projective k-planes contained in P and containing P0 = P(E0).

Deﬁnition 2.6. In general, we will refer to any B-scheme of the form Grassk(E) → B, for some vector bundle E → B, as a Grassmann bundle.

2.3.2. The relative Fano variety. Given a closed subscheme X ⊂ P = P(E), it is also possible to deﬁne the relative Fano variety of X → B,

Fk(X/B) = {(b, Λb) ∈ Grassk(P ) | Λb ⊂ Xb = X ∩ Eb}.

We show that Fk(X/B) is a variety if E and B are varieties. Let b be a point of the variety B (i. e. a closed point). Then we can find an affine neighborhood U of b such that −1 ∼ r r π (U) = U × P is trivial, and XU is cut out by homogeneous polynomials on P whose coefficients are regular functions on U (we can choose finitely many such polynomials, and a neighborhood of b on which all of their coefficients are regular). Then for each of these homogeneous polynomials, the relative Fano variety Fk(X/B)U is merely a form of the incidence correspondence “I” of Section 1.2.1. The argument given there shows that Fk(X/B)U is a closed subvariety of Grassk(E)U = U × G(k, r). (The case there was the universal family of hypersurfaces in a fixed projective space). Notice that we ignore the extra scheme-theoretic data from this definition: we consider the Fano variety as opposed to the Fano scheme. n Alternatively, if P ⊂ B × P is a projective sub-bundle over B and X ⊂ P a closed subvariety, then we may define the relative Fano variety simply as the subvariety of the n absolute Fano variety Fk(X) consisting of planes of the form {b} × Λ ⊂ B × P , i. e. those contained in the first ruling of the Segre embedding (as per [13] Theorem 9.22). In fact, the original “I” could also have been defined as such. 2.3.3. Some line bundles and Cartier divisors. We will interpret some basic objects and describe how they carry over to the context of projective bundles.

The sheaves OP (d). We will briefly describe how to extend the definition of the familiar n sheaf OP (d) to a general projective bundle P = P(E). The sections of the resulting sheaf FANO VARIETIES AND UNIRATIONALITY 15 ∼ OP (d)(d > 0) over an open subset B1 ⊂ B such that E|B1 = B1 × V are homogeneous forms of degree d on V with coefficients regular over B1. Thus we may conclude that d ∗ d ∗ OP (d)(PB1 ) = Sym E (B1), and therefore π∗(OP (d)) = Sym E . A global section of 6 OP (d) defines a closed subscheme of P → B. Let S ⊂ PV × V be the universal line bundle over PV = G(1,V )—for each ` ∈ PV,S r has fiber S` = ` ⊂ V. Define O(−1) = OP (−1) as the sheaf of sections of S, and define the “tautological sheaf” O(1) as the sheaf of sections of S∗ (see [15] vol. 2, ch. VI for a complete description). We may also define O(d) = O(1)⊗d. Similarly, define SP → P to be the universal line bundle over P = Grass0(E), i. e. the sub-bundle of P ×B E which has fibers (SP )(b,`b) = `b ⊂ Eb. We can now define ∗ OP (−1) to be the sheaf of sections of SP , OP (1) to be the sheaf of sections of SP , and ⊗d OP (d) = OP (1) . ∼ ∼ Now, let B0 ⊂ B be an affine open subset such that E|B0 = B0 ×V and P |B0 = B0 ×PV. We emulate the argument of [15] vol. 2, VI.1.4, Example 2, to find the sections of OP (1) over B × V. One can see from the definitions that S | = B × V × S, which 0 P P PB0 0 P PV implies S∗ | = B × V × S∗. Choose coordinates {X } on V, and consider the affine P PB0 0 P PV α P ∗ coordinate patches Uα = {Xα 6= 0} ⊂ PV. Exactly as in [15], the line bundle SP → PV is trivial over each open set B0 × Uα, with transition functions cαβ = Xα/Xβ between them. d Sections over B0 × Uα are regular functions: these are of the form ϕα = Pα/Xα with Pα a homogeneous form of degree d in the Xα with regular functions on B0 as coefficients. For d such a section to be regular on B0 × Uβ for all β, we must have cαβϕα = XαPα/(XβXα) ∗ regular on Uβ, which implies d = 1. Thus the sections of SP over PB0 = B0 × PV, i. e. the elements of OP (1)(B0 × PV ), are homogeneous linear forms with regular functions on B0 as coefficients. This description then clearly extends to non-affine open subsets B1 ⊂ B over which P is trivial. Likewise, the sections of OP (d) over B1 are forms of degree d. These homogeneous linear forms take values on E, so we may conclude that d ∗ π∗(OP (d)) = Sym E for d > 0. Cartier divisors. We will think of a Cartier divisor D on a scheme X as a global ∗ ∗ section of the sheaf MX /OX (defined in [17] II.6), which agrees exactly with the notion of a “locally principal divisor” on an irreducible variety in the classical setting. With this definition, a Cartier divisor D is effective iff it can be represented on an open cover {Uα} by elements of OX (Uα) ⊂ MX (Uα), called “local equations” for D. An effective Cartier divisor D thus defines a locally principal ideal sheaf ID and hence a closed subscheme of X, referred to by the same letter D. In [15] vol. 2, VI.1.4, classes of Cartier divisors are shown to be equivalent to invertible sheaves and line bundles, which allows us the pushforward and pullback operations, respectively (though one may lose local triviality in applying the pushforward). The Cartier divisors manifestly form a group. In the case of the projective bundle P = P(E), one can see from the previous description that the nonzero global sections of OP (d), for all d ≥ 0, form a graded submonoid of the 0 0 0 effective Cartier divisors. Furthermore, if α ∈ H (P, OP (d)) and β ∈ H (P, OP (d )) define 0 subschemes Pα ⊂ Pβ ⊂ P, respectively, then β/α will be a global section of OP (d − d).

6 Note that the vector bundles OP (d), d > 0, are not necessarily sub-bundles of trivial bundles, or even projectively embeddable; but this will remain true of all projective bundles we will consider, thereby not violating Remark 2.2. 16 AUTHOR: ALEX WALDRON ADVISOR: JOE HARRIS

To see this quickly, consider the subschemes of E deﬁned by α and β, meaning the cones Pˆα ⊂ Pˆβ over Pα ⊂ Pβ respectively. Over any aﬃne open set B0 ⊂ B as above, the homogeneous polynomials representing α and β are regular functions on E|B0 —so the scheme-theoretic inclusion Pˆα ⊂ Pˆβ implies that α| β as homogeneous polynomials on each B0 × PV. We will not, however, need the full facts of the case, i. e. that the group of line bundles OP (d) represents a subgroup of the class group, etc.

2.3.4. Residual varieties. The goal of this section is to formalize and generalize the “residual” intersections involved in the proof of the unirationality of cubic hypersurfaces, Propo- sition 2.3. Definition 2.7. A family of hypersurfaces over B is a closed subscheme X ⊂ P = P(E) defined by the vanishing of a global section sX of OP (d) that does not vanish identically over any point of B. By a family of l-planed hypersurfaces, we mean a triple of B-schemes (Γ,X,P ) such that Γ = P(E0) for a sub-bundle E0 ⊂ E, and Γ ⊂ X ⊂ P. By the previous discussion, 2.3.3, a family of hypersurfaces is an effective Cartier divisor on P. We define several projective bundles associated to the bundle P. Write

Π = Grassl+1(P ; Γ) for the family of projective (l + 1)-planes containing Γ (such a plane was referred to as n−l−1 Θ in the proof of Proposition 2.3). Π is then a P -bundle over B; it is in fact locally trivial, by Lemma 3.3 to follow. Let Π˜ ⊂ P ×B Π be the universal bundle over Π, whose ﬁber over a closed point of Π is the corresponding projective (l + 1)-plane in P. Consider the pullback bundles ΓΠ = Γ ×B Π, and PΠ = P ×B Π. Then we have the inclusions

ΓΠ ⊂ Π˜ ⊂ PΠ. n−l−1 Note that these are projective bundles over the base Π, itself a P -bundle over the original base B. Now, let X ⊂ P be a family of hypersurfaces over B, defined by a global section sX of OP (d). The pullback XΠ is again an effective Cartier divisor, in PΠ, defined again by sX (viewed as a section of OΠ(d)). Consider the scheme-theoretic intersection

XΠ ∩ Π˜ ⊂ Π˜.

This is yet again a Cartier divisor, defined by the restriction to Π˜ of sX which is in turn a ˜ section of OΠ˜ (d). This restriction does not vanish identically: for, Π is irreducible (being a vector bundle over a projective bundle over the irreducible variety B), and XΠ does not ˜ ∼ r contain any fiber Πb over b ∈ B (the latter projects surjectively onto Pb = P , while the former does not). Finally, note that the family of l-planes ΓΠ ⊂ Π˜ is an effective Cartier divisor, for which ˜ a defining section of OΠ˜ (1) can readily be described locally on Π. Definition 2.8. Corresponding to our given family of l-planed hypersurfaces (Γ,X,P ) over B, we have several constructions: (a) The main residual Π-scheme will refer to the effective Cartier divisor

Y := XΠ ∩ Π˜ − ΓΠ ⊂ Π˜. FANO VARIETIES AND UNIRATIONALITY 17

By the preceding discussion, Y is defined by a section of OΠ˜ (d − 1). (a’) Define the locus of definition of Y, DEF(Y ) ⊂ Π, to be the maximal open subset of (l + 1)-planes Θ ∈ Π for which YΘ ⊂ Π˜ Θ has dimension l, i. e. is a hypersurface. If DEF(Y ) is non-empty, then YDEF(Y ) → Π, will be called the residual family of hypersurfaces attached to (Γ,X,P ). (b) The secondary residual Π-scheme Z will refer to the scheme-theoretic intersection

Z := Y ∩ ΓΠ.

(b’) Write DEF(Z) ⊂ Π for the maximal open subset over which ZDEF(Z) → Π has fiber dimension l − 1, i. e. is a hypersurface. To give a very rough summary: the first of these, Y, is the family of intersections of X with the family Π of (l + 1)-planes containing the family of l-planes Γ ⊂ X, residual to Γ. The second, DEF(Y ) ⊂ Π, is the open subset having nontrivial fibers in Y. The third, Z, is the family of intersections of Y with the base family of l-planes. Notice that DEF(Z) ⊂ DEF(Y ). 2.4. Explicit description of the fibers of the residual varieties. Over an individual closed point b ∈ B, the fibers of the residual varieties are readily describable as subvarieties ∼ r of Pb = P . To do this, we work temporarily over the base B = Spec(K). Choose homo- r l geneous coordinates {V0,...,Vl,Wl+1,...,Wr} on P = P so that Γ = P = {Wi = 0}. n−l−1 Then {Wi} are homogeneous coordinates on Π = P . Write the defining equation of X(= Xb) as X I F (V,W ) = V FI (W ). 0≤|I|≤d−1

Here each I is a multi-index of degree |I|, and each FI (W ) is a homogeneous polynomial of degree d − |I| ≥ 1. Now, given a point [wl+1, . . . , wr] ∈ Π, we may parametrize the corresponding plane as

Θ = {[V0,...,Vl, wl+1U, . . . , wrU]}.

We thus have homogeneous coordinates [V0,...,Vl,U] on Θ such that Γ ⊂ Θ is the plane defined by U = 0. The restriction of F to the (l + 1)-plane Θ is given by X I d−|I| F (V,W )|Θ = V U FI (wl+1, . . . , wr). Since 0 ≤ |I| < d, we may divide through by U to obtain the defining equation for the primary residual Π-scheme Y, X d−|I|−1 I FY = F (V,W )|Θ/U = FI (wl+1, . . . , wr)U Z . The scheme-theoretic intersection of Y with Γ is given simply by setting U = 0, by which we obtain the defining equation for the secondary residual scheme Z, X I0 FZ (V, Θ) = FI0 (wl+1, . . . , wr)Z . |I0|=d−1

Here, the FI0 are homogeneous linear forms. From these equations, we see that the locus n−l−1 P − DEF(Y ), i. e. the set of Θ which are equal to YΘ, is the common zero locus 18 AUTHOR: ALEX WALDRON ADVISOR: JOE HARRIS

n−l−1 of the homogeneous forms FI . Likewise, the locus P − DEF(Z) is the common zero locus of the linear forms FI0 and thus a subset of the former; hence DEF(Z) ⊂ DEF(Y ). From this discussion, we conclude the following:

Proposition 2.9. Let (Γ,X,P ) be an l-planed family of hypersurfaces over an integral base B, and let b ∈ B be a closed point. Then,

(a) If Γb meets the smooth locus of the morphism π : X → B (i. e. the smooth locus of Xb), then DEF(Z) ∩ Πb 6= ∅. p (b) Let P ⊂ Πb be the common zero locus of the linear forms FI0 coming from the terms of FZ . Then the secondary residual schemes

{ZΘb }

∼ l form a linear system of hypersurfaces in Γb = P parametrized by the quotient of Πb by the p r−p−1 subspace P , i. e. P . The basepoint locus of this system is contained in the singular locus of the morphism π : X → B.

Remark 2.10. For our purposes, the “smooth locus” of the map π is the dense open subset of X on which dim(ker(dπ)) attains a Zariski-local minimum (see [13], ch. 14). So, by Proposition 3.2 (to follow), a closed point x ∈ X lies in the smooth locus of π if and only if the scheme-theoretic ﬁber Xπ(x) through x is smooth at x. Thus the requirement of Proposition 2.9 is simply that for each point b ∈ B, Γb not be contained in the singular locus of the hypersurface Xb ⊂ Pb.

Proof. Assume that the constructions of this section have been made for the given closed point b ∈ B, i. e. F (V,W ) deﬁnes Xb ⊂ Pb, the Vi are coordinates on Γb, and so-on.

(a) By the remark, we know that Γb 6⊂ (Xb)sing. The V -partials of F (V,W ) are all zero on Γb = {Wi = 0}. A W -partial along Γb is

∂F (V,W ) X I0 = V FI0 (0,..., 1,..., 0) = FZ (V, 0,..., 1,..., 0), ∂Wi Γb |I0|=d−1 with a 1 in the i’th argument of each. So, ∂F (V, 0) 6= 0 ⇒ F (V, 0,..., 1,..., 0) 6= 0 ⇒ ∂Wj Z [0,..., 1,..., 0] ∈ DEF(Z)b. Hence (Xb)sm 6= ∅ ⇒ DEF(Z)b 6= ∅

(b) By the deﬁnition of FZ , the hypersurfaces ZΘb manifestly form a linear system of p 0 degree d − 1 in Γb parametrized by the quotient Πb/P . A basepoint (V , 0) of this system would yield a solution, for all W, to the equation

0 X I0 X ∂F 0 0 = FZ (V ,W ) = V FI0 (W ) = Wi (V , 0). ∂Wi |I0|=d−1

This would imply ∂F (V 0, 0) = 0 for all i, meaning that V 0 is a singular point of X . Thus ∂Wi b the basepoint locus of {Zθb } is contained in the singular locus of Xb. FANO VARIETIES AND UNIRATIONALITY 19

2.5. Numbers. Finally, we introduce the relevant numerical bounds of which we will make use in the following section. Deﬁne

l + d − 1 M(d, l) = + l − 1. d − 1 This number has the following use: Lemma 2.11. Let d, l ≥ k ≥ 0, and n ≥ M(d, l) be non-negative integers, and let n Λ ⊂ X ⊂ P be a k-plane contained in a hypersurface X of degree d. Then there exists an l-plane Γ such that Λ ⊂ Γ ⊂ X. Proof. Since M(d, l0) ≤ M(d, l) for all l0 ≤ l, it is suﬃcient to assume k = l − 1. Choose n homogeneous coordinates V0,...,Vk,Wk+1,...,Wn on P such that the k-plane Λ = {Wi = 0}. In these coordinates, the deﬁning polynomial of X is of the form X I F (V,W ) = V FI (Wk+1,...,Wn). 0≤|I|

Next we fix an integer k ≥ 0 and define two functions recursively, N0(d, k) and N(d, k). Begin by setting k + 1 N(2, k) = N (2, k) = + 3. 0 2 Then for d ≥ 3, define recursively

N0(d, k) = M(d, N(d − 1, k) + 1) and then k + d N(d, k) = N (d, k) + + 2, 0 d or in other words, k + d − 1 N (d, k) = M(d, N (d − 1, k) + + 3). 0 0 d − 1 Note that both of these functions are strictly increasing with k.

Proposition 2.12. (a) In the case d = 2, note that N0(2, k) ≥ M(2, k). So for n ≥ n N0(2, k). So a quadric hypersurface Q ⊂ P is swept out by k-planes. This in turn implies that if Q is smooth, then dim(Fk(Q)) = φ(n, 2, k). (b) For all d ≥ 3 and k ≥ 0, k + d N (d, k) ≥ + 3k + 1 ≥ M(d, k), 0 d 20 AUTHOR: ALEX WALDRON ADVISOR: JOE HARRIS

n and therefore a hypersurface X ⊂ P with n ≥ N0(d, k) is swept out by k-planes. k+1 Proof. (a) For d = 2, note that M(2, k) = k + 1 + k − 1 = 2k ≤ 2 k + 3, which n is the first claim. So by Lemma 2.11, any quadric in P , n ≥ N0(d, k), is swept out N by k-planes. In particular, this implies that the map π1 : I → P from Chapter 1 is surjective: so for a general quadric hypersurface Q, the arguments of Chapter 1 imply N that dim(Fk(Q)) = dim(I) − dim(P ) = φ(n, 2, k). However, a general quadric is smooth, and all smooth quadrics are projectively equivalent, so this applies to all smooth quadrics. (b) Fix k. We first do the base case d = 3 : k+1 + 3 + k+2 + 2 + 3 (k + 1)2 + 8 k + 3 N (3, k) = 2 2 ≥ ≥ + 3k + 1 0 2 2 3 by comparing coefficients. Now, for a given d ≥ 3 and k > 1, assume the result for all (d0, k0) such that d0 < d or k0 < k. Then, k + d − 1 2(k + d − 1) k + d − 1 N (d, k) ≥ M(d, 2 + 3k + 4) ≥ (2 + 3k + 4 − 1) 0 d − 1 d − 1 d − 1 k + d ≥ 2 + 3k + 3. d 2.6. Low-degree smooth hypersurfaces do not have too many k-planes.

Theorem 2.13. Let n, d, k be positive integers, and let N0 = N0(d, k) as deﬁned above. n If n ≥ N0(d, k) and X ⊂ P is a smooth hypersurface, then

dim(Fk(X)) = φ(n, d, k). We will need two corollaries during the induction step of the proof. (In the next chapter, we will in fact need two further corollaries to this theorem; but these will be deferred until section 3.4.2.)

Corollary 2.14. (a) If the codimension of Xsing in the hypersurface X is at least N0, then we may likewise conclude

dim(Fk(X)) = φ(n, d, k).

n (b) Let N(d, k) as deﬁned earlier, and let X ⊂ P be any hypersurface of degree d. If n ≥ N(d, k), then

dim(Fk(X)) ≤ max{φ(n, d, k), φ(n, d, k) + dim(Xsing) − 1}.

Proof of Corollary 2.14. (a) Assume the theorem for a given d and k. For n < N0 the desired corollary is vacuous, and for n = N0 it is just the theorem. We will prove the n cases n > N0 by induction on n—so let n > N0 and X ⊂ P be a hypersurface of degree ∼ n−1 n d whose singular locus has codimension at least N0. Let H = P ⊂ P be a general ∼ hyperplane, and consider G(k, H) = G(k, n − 1) ⊂ G(k, n), the sub-Grassmannian of k- planes contained in H. Set Y = X ∩ H. Then by Bertini’s Theorem, the singular locus of k+d Y again has codimension at least N0, so by induction codim(Fk(Y ) ⊂ G(k, n)) = d . But Fk(Y ) = Fk(X) ∩ G(k, H) ⊂ G(k, n); so, by the subadditivity of codimension of FANO VARIETIES AND UNIRATIONALITY 21 intersections in the smooth variety G(k, n) ([13] 17.24), if a component F of Fk(X) meets G(k, H) then (6) k + d codim(F ⊂ (k, n)) ≥ codim(F (X) ⊂ (k, n)) ≥ codim(F (Y ) ⊂ (k, H)) = . G k G k G d

It remains to show that every component of Fk(X) meets G(k, H). This will follow from a standard result in the intersection theory of Grassmannians, which we state without proof:

Lemma 2.15. If X1 and X2 are subvarieties of G(k, n) of codimension c1 and c2 such that c1 + c2 < n + 1 − 2k, then the intersection X1 ∩ X2 is nonempty. n Note that in the case k = 0, this gives the standard intersection criterion on P . k+d Now, recall that Fk(X) is cut out by d conditions, hence any component F0 has codimension at most that number. So the codimensions of F0 and G(k, H) ⊂ G(k, n) are k+d at most d and k + 1, respectively. We have k + d k + d n + 1 − 2k ≥ N + 1 − 2k ≥ + 3k + 2 − 2k = + k + 2 0 d d by Proposition 2.12(b), and so by the lemma F meets G(k, H), which establishes 6 and the claim. (b) Assume the result of part (a) for the given pair d, k, and let X be a hypersurface of n degree d in P , with n ≥ N(d, k). If codim(Xsing ⊂ X) ≥ N0 then part (a) applies; otherwise dim(Xsing) ≥ n − N0 ≥ k+d N(d, k) − N0 = d + 2, by deﬁnition of N(d, k). So, dim(Fk(X)) ≤ G(k, n) k + d = φ(n, d, k) + d ≤ φ(n, d, k) + dim(Xsing) − 1. Remark 2.16. For any given n, d, k, these corollaries can be deduced from the theorem as applied only to these n, d, k. Thus we will be able to use the corollaries in conjunction with our induction hypothesis.

2.6.1. A Bertini Lemma. We will need the following lemma to control the singularities of our hypersurfaces.

m Lemma 2.17. (Bertini’s Theorem for a projective space) Let D = {Dα}α∈P be a linear n n system of hypersurfaces in P , with base locus B ⊂ P of dimension b. Deﬁne the subsets Sk = {Dα ∈ D | dim((Dα)sing) ≥ b + k}.

Then Sk is a projective variety and m codim(Sk ⊂ P ) ≥ k. 22 AUTHOR: ALEX WALDRON ADVISOR: JOE HARRIS

Proof. Deﬁne the incidence correspondence m n Σ = {(α, p) ∈ P × P | p ∈ (Dα)sing}. m −1 m Then Sk = {α ∈ P | dim(π1 (α)) ≥ k + b}, which is a projective variety in P since m ﬁber dimension is an upper-semicontinuous function of α ∈ P (Proposition 1.3(b)). We may restrict π1 to be surjective onto Sk, to obtain the relation −1 dim(Sk) + k + b = dim(π1 (Sk)) ≤ dim(Σ), so we will be done upon establishing (7) dim(Σ) ≤ m + b.

We will examine the projection π2 to obtain the bound in (7) on dim(Σ). Consider the n m ∗ map f : P − B → (P ) associated to the linear system D. Explicitly, let the vector m space of meromorphic functions underlying P have basis {f0, . . . , fm}, so we may write f(p) = [f0(p), . . . , fm(p)] . Since p is outside the base locus B, we may assume f0(p) 6= 0 so we can take aﬃne coordinates z1, . . . , zm with which to write

∂(fi/f0) (dfp)ij = . ∂zj p Conveniently, the rank l of this matrix is by definition the number of linear conditions for a hypersurface Dα 3 p corresponding to α = [a0f0 + ... + amfm] to be singular at the point p. Thus if rk(dfp) = l then the space of hypersurfaces singular at p, which is the −1 ∼ m−l−1 m inverse image of p under π2, is a linear subspace π2 (p) = P ⊂ P . n Define the locally closed subset Wl = {p ∈ P − B | rk(dfp) = l}. We have just determined that the fiber of π2 over a point of Wl has dimension m − l − 1, so −1 dim(Wl) + (m − l − 1) ≥ dim(π (Wl)). −1 −1 Since Σ is a union of the finitely many inverse images π2 (Wl) along with π2 (B), we must show that

m + b ≥ dim(Wl) + m − l − 1(8)

(9) b + l + 1 ≥ dim(Wl). We will in fact be done once we show this inequality, because (trivially) the inverse image −1 π2 (B) has dimension at most m + b. We have translated a question concerning the singular loci (Dα)sing into a question of the rank of dfp in order to apply Sard’s theorem. This result ([13] Proposition 14.4) states that over a field of characteristic 0, dfp is surjective for points landing in a nonempty open subset of f(Wl); so clearly dim(f(Wl)) ≤ l. This now implies −1 dim(Wl) ≤ dim(f (f(p))) + l, for any fiber through a point p ∈ Wl. But the fibers of f all have dimension less than b+1. In fact, any non-constant map on the complement of a b-dimensional variety of projective space has fiber dimension at most b + 1, by the following argument: FANO VARIETIES AND UNIRATIONALITY 23

Write f = [F0,...,Fm] so that the Fi are homogeneous polynomials with no common factor which are simultaneously zero nowhere outside of B (See [15] vol. I ch. 3). The projective variety defined by the Fi has dimension at most b. The fiber over (say) p = [1, 0,..., 0] then is defined by F1 = ··· = Fm = 0. The section of this fiber by the hypersurface {F0 = 0} has dimension one less (since the Fi have no common factor), and is contained in B. So the fiber over p, defined by F1 = ··· = Fm = 0, has dimension at most b + 1. This establishes (9), which implies (7) and hence the main inequality of the lemma. 2.6.2. Proof of Theorem 2.13. Theorem 1.6(c) of Chapter 1 asserts that in case φ(n, d, k) > 0 and d ≥ 3, every hypersurface contains a family of k-planes of dimension at least φ (the proof invoked the result of [1]). So it remains to show that a smooth hypersurface of sufficiently high dimension has a family of k-planes of dimension at most φ. To this end, we will construct a convenient family of pairs of planes, ∆. We will then find a lower bound L (step 1) and an upper bound U (step 2) on dim(∆). The resulting inequality L ≤ U will simplify to the desired upper bound on dim(Fk(X)) (step 3). The bounds L and U will be established by induction on d. Proposition 2.12(a) exactly establishes the base case d = 2 of the theorem, for all k. For the induction (the remainder of the proof), fix d and k, and assume that Theorem 2.13 and its corollaries hold for all triples (n0, d0, k0) with n0 ≤ n, d0 ≤ d, k0 ≤ k but not all equal. Note that the theorem is 0 0 0 n vacuous for n < N0(d , k ). Let X ⊂ P , with n ≥ N0(d, k), be a smooth hypersurface of degree d. Set l = N(d − 1, k), and define the locally closed subvariety ∆ ⊂ Fk(X) × Fl(X) given by ∆ = {(Λ, Γ) ∈ Fk(X) × Fl(X) | dim(Λ ∩ Γ) = k − 1}. Equivalently, ∆ can be defined by the condition that Λ and Γ together span an (l+1)-plane n in P (not necessarily contained in X). Remark 2.18. The idea is roughly as follows: our numbers are sufficiently high that not only will X be swept out by k-planes, but each of these k-planes in X is contained in an l = N(d − 1, k)-plane in X. This will allow us to apply our induction hypothesis on d first to show that ∆ is dense in a fiber product of flag-Fano varieties (step 1), then to inject ∆ into the relative Fano variety of the main residual scheme Y coming from the intersection of X with the variety of (N(d − 1, k) + 1)-planes containing its Fano variety of l = N(d − 1, k)-planes (step 2).

Step 1. Let Fk−1,k(X) and Fk−1,l(X) be the ﬂag-Fano varieties

Fk−1,k(X) = {(Ω, Λ) | Ω ⊂ Λ ⊂ X} ⊂ G(k − 1, n) × G(k, n) and Fk−1,k(X) = {(Ω, Γ) | Ω ⊂ Γ ⊂ X} ⊂ G(k − 1, n) × G(l, n). We will realize ∆ as an open, dense subset of the ﬁber product

Φ := Fk−1,k(X) ×Fk−1(X) Fk−1,l(X) From the deﬁnition, there is a regular map

∆ → Fk−1(X) 24 AUTHOR: ALEX WALDRON ADVISOR: JOE HARRIS given by (Λ, Γ) → Ω := Λ ∩ Γ. Via this map, embed ∆ ⊂ Φ via the identification (Λ, Γ) 7→ ((Ω, Λ), (Ω, Γ)) ∈ Φ. This embedding is open, since ∆ ⊂ Φ is defined by the open condition Λ 6⊂ Γ. We claim that ∆ is dense in the fiber product Φ. A point of Φ − ∆ can be represented by a triple (Ω ⊂ Λ ⊂ Γ). Since n ≥ N0(d, k) = M(d, N(d − 1, k) + 1) = M(d, l + 1), we can choose an (l + 1)-plane Σ ⊂ X such that Γ ⊂ Σ. Then the k-plane Λ ⊂ Γ can be realized as a limit of k-planes Λx 6⊂ Γ—simply choose a line ` that meets Γ in a single point x0 ∈ Λ − Ω, and set Λx = Λ, x, for points x ∈ `. Thus dim(∆) = dim(Φ). We get a standard lower bound on dim(Φ). Note that the fibers of the second projection k∗ Fk−1,k → G(k, n) are isomorphic to P , so

dim(Fk−1,k(X)) = k + dim(Fk(X)).

Similarly, the ﬁbers of Fk−1,l → G(l, n) are equal to G(k − 1, l), so we have

dim(Fk−1,l(X)) = k(l − k + 1) + dim(Fl(X)).

We have assumed that n ≥ N0(d, k) ≥ M(d, l) ≥ M(d, k), so we know that both Fk−1,k(X) and Fk−1,l(X) map surjectively onto Fk−1(X). So by [13] 11.15,

dim(∆) = dim(Φ) ≥ dim(Fk−1,k(X)) + dim(Fk−1,l(X)) − dim(Fk−1(X)).

= k + dim(Fk(X)) + k(l − k + 1) + dim(Fl(X)) − dim(Fk−1(X)). (Note: From the proof of this last fact, we have equality if ∆ is irreducible.) Since N0(d, k) ≥ N0(d, k − 1), we have from the induction hypothesis k + d − 1 dim(F (X)) = φ(n, d, k − 1) = k(n − k + 1) − , k−1 k − 1 and the last inequality reduces to k + d − 1 dim(∆) = dim(F (X)) ≥ k(l − n + 1) + + dim(F (X)) + dim(F (X)) =: L k−1 k − 1 k l

Step 2. The idea here is to get an upper bound U on dim(∆) in terms of just dim(Fl(X)), so that the expression L ≤ U will involve only dim(Fk(X)) as an unknown quantity. n We will regard the ambient space P = P as a projective bundle over SpecK. Let F = Fl(X), and let F˜ → F be the universal projective bundle, whose fiber over a point Γ ∈ F is the corresponding projective l-plane. Consider the (rather trivial) family of l-planed hypersurfaces (F,X˜ F ,PF ). Form the n−l−1 P -bundle Π → F corresponding to this family—recall from section 2.8 that the fiber of Π over a point Γ ∈ F is the set of (l +1)-planes Θ containing Γ. Also form the main and secondary residual Π-schemes, respectively Y ⊃ Z. Recall that the relative Fano variety ˜ Fk(Y/Π) ⊂ Grassk(Π) is the variety of pairs (Λ, ΘΓ) such that Λ ⊂ YΘΓ ), where Λ is a k-plane and ΘΓ ⊃ Γ an (l + 1)-plane. Define the map of varieties

∆ → Fk(Y/Π)

(Λ, Γ) 7→ (Λ, ΘΓ), FANO VARIETIES AND UNIRATIONALITY 25

where ΘΓ = Λ, Γ is the (l + 1)-plane they span. Note that Λ ⊂ YΘΓ as required: the open dense set Λ − Γ ⊂ Λ is certainly contained in the residual variety YΘΓ , which is closed and hence contains Λ. This map is clearly injective. So in what follows, we will simply compute an upper bound on the dimension of Fk(Y/Π). This will furnish our upper bound U on the dimension of ∆. To do this, ﬁx an l-plane Γ0 ⊂ X, and we will compute the dimension of the ﬁber of Fk(Y/Π) over Γ0. For this purpose, we need only consider the residual family over the single l-plane Γ0, i. e. n−l−1 Y0 := {YΘ | Θ ∈ ΠΓ0 = P }. So we would like to compute the dimension of

Fk(Y )Γ = Fk(Y ) = Fk(Y0 n−l−1 ). /Π 0 /ΠΓ0 /P

The YΘ have degree d − 1, so we hope to apply the induction hypothesis once we know something about their singularities. Happily, since X is smooth, by Proposition 2.9 the hypersurfaces {ZΘ = YΘ ∩ Γ0} form a base-point-free linear series in Γ0—and so by Proposition 2.17, the locus n−l−1 Sρ = {Θ ∈ DEF(Z)Γ0 ⊂ P | dim((ZΘ)sing ≥ ρ − 1} n−l−1 has codimension at least ρ in P . Hence, the variety

Wρ = {Θ ∈ DEF(Z)Γ0 ⊂ DEF(Y )Γ0 | dim((YΘ)sing ≥ ρ} ⊂ Sρ n−l−1 must also have codimension at least ρ in P .

Thus we can choose Θ ∈ DEF(Z)Γ0 −W1, and the hypersurface YΘ has at most isolated singularities. So, since l = N(d − 1, k), the induction hypothesis yields from Corollary 2.14 that

dim(Fk(YΘ)) = φ(l + 1, d − 1, k).

We may thus conclude that the inverse image of DEF(Z) in the fiber F (Y n−l−1 ) = k 0/P Fk(Y/Π)Γ0 has dimension φ(l + 1, d − 1, k) + n − l − 1. It remains to check that the inverse image of the p-dimensional projective linear space c n−l−1 (DEF(Z)Γ0 ) = P − DEF(Z)Γ0 (as per Proposition 2.9) does not introduce components of larger dimension in Fk(Y/Π)Γ0 . Since the series {ZΘ} is base-point free, we must c n−l−1 have l + 1 ≤ codim((DEF(Z)Γ0 ) ⊂ P ), since this is the dimension of the linear system {ZΘ}. Meanwhile we have the trivial upper bound dim(Fk(YΘ)) ≤ dim(G(k, l + 1)), so the fiber dimension can jump from the generic dimension by at most dim(G(k, l + 1)) − k+d−1 c φ(l + 1, d − 1, k) = d−1 . But since the locus (DEF(Z)Γ0 ) having possibly higher- k+d−1 dimensional fibers has codimension at least l + 1 ≥ d−1 , these fibers cannot contribute a higher-dimensional component of Fk(Y/Π)Γ0 . We can conclude that

dim(Fk(Y/Π)Γ0 ) = φ(l + 1, d − 1, k) + n − l − 1.

Since Γ0 ∈ Fl(X) was arbitrary, we may conclude ﬁnally that

dim(∆) ≤ dim(Fk(Y/Π)) = dim(Fl(X)) + φ(l + 1, d − 1, k) + n − l − 1 k+d−1 = dim(Fl(X)) + (k + 1)(l − k − 1) − d−1 + n − l − 1 =: U 26 AUTHOR: ALEX WALDRON ADVISOR: JOE HARRIS

Step 3. We now simplify the inequality L ≤ dim(∆) ≤ U from steps 1 and 2: k + d − 1 k(l − n + 1) + + dim(F (X)) + dim(F (X)) k − 1 k l k + d − 1 ≤ dim(F (X)) + (k + 1)(l − k − 1) − + n − l − 1. l k

Cancelling Fl(X) and rearranging, we get

dim(Fk(X)) ≤ (k + 1)(l − k + 1) − k(l − n + 1) + n − l − 1 k+d−1 k+d−1 − k−1 − k k+d = (k + 1)(l − k − 1 − (l − n + 1)) − d k+d = (k + 1)(n − k) − d . This establishes the desired upper bound dim(Fk(X)) ≤ φ, which is equal to the lower bound from Chapter 1, yielding the desired equality.

Remark 2.19. Since this last upper bound on dim(Fk(X)) is sharp, the equivalent inequality L ≤ U of the bounds on dim(∆) must also be an equality, i. e. L = dim(∆) = U. So the sharpness of our upper bound on dim(Fk(X) reflects the fact that the arguments of steps 1 and 2 were each an accurate computation of the dimension of ∆. Also note that since X in the proof is swept out by k-planes, Theorem 1.6(c) is trivial, as argued prior to Remark 1.9. So the full result of [1] is not required for Theorem 2.13. 3. Unirationality of Smooth Hypersurfaces of Low Degree. In this chapter we will demonstrate an important consequence of Theorem 2.13: the unirationality of smooth, low-degree hypersurfaces. Our main reference will again be [2]. In Section 3.3, however, we arrive at similar constructions to those of [2] much more straightforwardly. The methods of this chapter were originally developed for the case of a general hypersurfaces. The conclusion of the last chapter will be used to prove Corollary 3.12: we are able to bound the dimension of the variety of planes contained in the smooth fibers in order to “inflate” the image, showing surjectivity. Note: All fibers of morphisms will now be “scheme-theoretic,” as will all inclusions. A “point,” however, will still refer always to a closed point. Example 3.1. Our method of proof in this chapter generalizes that of Proposition 2.3 (a review of this example is recommended). We begin with a heuristic attempt to prove the case of quartic hypersurfaces, identifying the main obstructions to its success. n Let X ⊂ P be a smooth quartic hypersurface. Hoping to extend the proof of Propo- sition 2.3, we take n sufficiently large that we can choose an l-plane Γ ⊂ X and consider the (l + 1)-planes Θ that contain it. A general such plane Θ intersects X in the union of a residual cubic hypersurface YΘ ⊂ Θ and the plane Γ. As before, the blow-up ˜ n−l−1 πΓ : X = BlΓ(X) −→ P = Π l+1 has general fiber a cubic hypersurface YΘ ⊂ P . 7 Now, we expect that a general fiber YΘ of X˜ → Π will be smooth. In this case, YΘ will be unirational, by Proposition 2.3. However, if we hope to use this fact, we must be able to choose a k-plane ΩΘ on each cubic YΘ. As before, we can narrow down the choice by looking only within the l-plane Γ, which meets YΘ in the secondary residual scheme ZΘ = Γ ∩ YΘ. As in the case of cubics, we do not expect to choose a unique such plane for each ZΘ, i. e. a rational section. Instead we set up the incidence correspondence n−l−1 Ψ := {(Θ, Ω) | Ω ⊂ ZΘ = YΘ ∩ Γ} ⊂ P × G(k, Γ). (For cubics, we had k = 0 so that G(0, Γ) = Γ.) As before, we form the pullback of n−l−1 X˜ → Π = P to this variety Ψ → Π,

H := X˜ ×Π Ψ.

The family H1 → Ψ is then a k-planed family of cubic hypersurfaces. This gives us a unirational parametrization of each cubic YΘ. And we should be able to further pull back H1 to obtain a “pointed, k-planed family of hypersurfaces” H0, dominated by a variety which will be rational—provided that Ψ itself is rational. (See Section 3.4.1 for a better description.) The question of whether Ψ is rational can be approached as before, by considering the projection map to the second factor G(k, Γ). As in Proposition 2.9, the secondary residual hypersurfaces {ZΘ} form a linear system, and each point Ω ⊂ Γ imposes a certain number of linear conditions on Θ ∈ Π. Thus we may hope that Ψ is a projective bundle over the rational base G(k, Γ), and will hence be rational.

7In fact it will be suﬃcient that the general ﬁber have at worst isolated singularieties when n d, as in the proof of Theorem 2.13. 27 28 AUTHOR: ALEX WALDRON ADVISOR: JOE HARRIS

There are two main obstructions to these last statements: first, the variety Ψ may not dominate G(k, Γ)—then its image might not be rational, and neither might Ψ. However, we should be able to fix this by choosing n l, so that a k-plane Ω imposes fewer than n − l − 1 conditions on the parameters Θ ∈ Π of the linear system {ZΘ}—then, each Ω is contained in some hypersurface ZΘ. Also note that although each fiber of Ψ → G(k, Γ) is a linear space, they may not all have the same dimension—nor do all k-planes necessarily impose the same number of conditions on the series {ZΘ}. Thus the variety Ψ may be reducible, which is likewise true of H. However, Corollary 3.12(a) will assert that for n appropriately large, a general k-plane imposes independent conditions on a linear series n in P . The second difficulty is that even if a component of Ψ0 ⊂ Ψ dominates G(k, n), and is therefore rational, we cannot be sure that Ψ0 also dominates the other factor Π (and vice versa). If Ψ0 does not dominate Π, then H0 = X˜ ×Π Ψ0 cannot dominate X˜ as required. However, this will be remedied in what follows by Corollary 3.12(b) to Theorem 2.13, which asserts that for n d, every component of Ψ dominates Π. Both of these failures in surjectivity, as well as the reducibility of Ψ, do in fact occur for incidence correspondences Ψ attached to certain linear series: examples are described in [2]. For the case of quartics, we will show that for n ≥ U(4) = 179124155 8 n (as defined in Corollary 3.8), these problems do not occur and a smooth quartic in P will be unirational.

3.1. Preparatory results, and combs. For the sake of completeness, we will prove two necessary basic facts. Lemma 3.2. Let Φ: X → Y be a map of Noetherian K-schemes, and y ∈ Y a closed point. Let x ∈ Xy = X ×Y Spec(κ(y)) = X ×Y K be a closed point of the ﬁber over y (coming from a closed point of x), and let dΦx : TxX → TyY be the diﬀerential. Then the sequence of K-vector spaces

dΦx 0 −→ TxXy −→ TxX −→ TyY associated to the morphisms Xy → X → Y is exact. Proof. Since the statement is local, we may set X = Spec(S) and Y = Spec(T ). Let φ : T → S be the ring homomorphism corresponding to Φ. Let m ⊂ S be the maximal ideal corresponding to the point x, and let n ⊂ T be the maximal ideal corresponding to y. The given Φ(x) = y means just that φ−1(m) = n, so (also by deﬁnition) there is an induced homomorphism φx : Tn → Sm of the local rings such that φx(nn) ⊂ mm. The ﬁber product X ×Y Spec(κ(y)) is the spectrum of the ring

S ⊗T K = S ⊗T Tn/nn = S ⊗T T/n.

8As mentioned in the introduction, this number is slighly lower than it should be due to a typo in the original paper. FANO VARIETIES AND UNIRATIONALITY 29

(The fact T/n = Tn/nn is [19] Prop. 5.1.5.) The inverse image “x” of x in the ﬁber Xy corresponds to the ideal (m ⊗ 1)(S ⊗T K) = m ⊗T K. Furthermore, the local ring (S ⊗T

K)m⊗T K is naturally isomorphic to Sm ⊗T K, under which the maximal ideal corresponds to mm ⊗T K—this is readily seen from examining the surjective ring homomorphism S → S ⊗T 1, which expresses S ⊗T K ≈ S/(φ(n)m), and from the exactness of localization ([16] Ch. 3). We prove the exactness of the dual sequence to the one stated,

∗ ∗ ∗ Ty Y −→ Tx X −→ Tx Xb −→ 0(10) 2 φx 2 2 (11) nn/nn −→ mm/mm −→ (mm/mm) ⊗T K −→ 0. We may write out

2 2 2 (mm/mm) ⊗T K = (mm ⊗T T )/(mm ⊗T T + mm ⊗T n) ≈ mm/(mm + φ(n)mm).

But φ(n)mm = φx(nn)mm in the local ring OX,x = Sm since any element s 6∈ n is sent to 2 φ(s) 6∈ m. Thus the right side of the last equation is equal simply to (mm/mm)/(φx(nn)), which is just the desired statement that (10) is exact. n Proposition 3.3. Let X ⊂ B × P be a family, with X and B varieties. If the scheme- n theoretic ﬁbers Xb over closed points b ∈ B are k-dimensional linear subspaces of P , then X is a projective bundle of rank k over B, i. e. is locally trivial.

n Note. The requirement that X be a sub-bundle of a trivial bundle, i. e. X ⊂ B × P , is essential, as per Remark 2.2.

∼ k Proof. Given b0 ∈ B a closed point and Xb0 = P a ﬁber, choose an (n − k − 1)-plane ∼ n−k−1 n Λ = P ⊂ P such that Xb0 ∩ Λ = ∅. Let U be the open subset {b ∈ B | Xb ∩ Λ = ∅}. n k Let πΛ : P − Λ → P be the projection from Λ. Then Id × πΛ is bijective between the −1 k closed points of XU = π1 (U) and U × P (linear algebra). From [13] Corollary 14.10, it remains to show that Id × πΛ has injective diﬀerential at all closed points (b, p) ∈ XU . Form the diagram

0 / T(b,p)Xb / T(b,p)XU / TbU / 0 ,

(Id×πΛ)(b,p) / k / k / // 0 T(b,πΛ(p))({b} × P ) T(b,πΛ(p))(U × P ) TbU 0 where (if this is not obvious) the left vertical map is derived from the sequence

k ∼ k k k P = Xb → XU → U × P → P → {b} × P , which is an isomorphism since nonconstant. The left and right vertical maps in the diagram are thus isomorphisms, therefore the central map is an isomorphism. Alternatively, notice that [14], III-56 directly implies that X → B is a ﬂat family. So X is a ﬁber product with the Grassmannian bundle, hence a bundle itself. 30 AUTHOR: ALEX WALDRON ADVISOR: JOE HARRIS

3.1.1. Combs. The following is a key object in our constructions: Deﬁnition 3.4. A comb over B is a B-scheme C for which there exists a sequence

G G ··· Gn−1 G 1 2 n J HH AA HH JJ S HH S A HH S J H AA HH JJ HH AA HH JJ HH# A H# JJ% C = C0 / C1 / C2 ··· Cn−1 / Cn = B, where each Gi is a Grassmann bundle over Ci, and Ci ⊂ Gi+1 is a dense open subset. By standard arguments, each of these maps is dominant. We will say that a map C → B for which there exists such a diagram is a comb morphism. Clearly, comb morphisms are closed under composition. Proposition 3.5. If C → B is a comb and B is rational, then C is rational. Proof. A Grassmann bundle over B is birational to B×G for some ordinary Grassmannian G. Grassmannians are rational by Proposition 1.1. Hence Cn−1 is rational, and so-on. 3.2. Statement of results concerning unirationality. We first must define another lower bound. Define, inductively, L(2) = 0, and L(d) = N(d, L(d − 1)) for d ≥ 3. Theorem 3.6. Let (Γ,X,P ) be an l-planed family of hypersurfaces over an integral base B, with l ≥ L(d). Assume Γ is contained in the smooth locus of the morphism π : X → B. Then there exists a comb D → B, and a morphism D → X such that for each closed point b ∈ B, the map fb : Db → Xb is dominant. The proof of this theorem occupies the remainder of the paper. Note that in this case, the map f will be referred to as “fiber-by-fiber” dominant. Recall that as in Remark 2.10, the smoothness requirement here is simply that fiber Xb be smooth at all points of the plane Γb ⊂ Xb. Corollary 3.7. If B is rational in Theorem 3.6, then X is unirational. Proof. From Corollary 3.5, the comb D is rational. In the theorem, D → X is dominant therefore X is unirational. Corollary 3.8. For 1 L(d) + d n ≥ + L(d) =: U(d), L(d) + 1 d n a smooth hypersurface X ⊂ P of degree d is unirational. Proof. The requirement here is that φ(n, d, L(d)) ≥ 0, so that X contains an L(d)-plane (Ch. 1 Theorem 1.6). Remark 3.9. The arguments and results of this chapter, particularly Theorem 3.6, also apply to the case of a non-algebraically-closed field k of characteristic zero. However this is not true of the results of Chapter 2, and therefore Corollary 3.8 fails as stated for arbitrary k. Still, the connection between Fano varieties and unirationality persists: FANO VARIETIES AND UNIRATIONALITY 31

n Corollary 3.10. If a smooth hypersurface X ⊂ Pk of degree d contains an L(d)-plane that is rational over k, then X is unirational over k. As an example of how this might work, consider the results of Section 3.1 above. Note that the L(d)-plane Λ ⊂ X is rational over k if and only if the residue ﬁeld at each closed point x ∈ L(d) is equal to k—and in this case, the tangent spaces at x and Φ(x) are ﬁnite-dimensional k-vector-spaces, and the statement (and proof) of Lemma 3.2 makes sense.

3.3. Constructions. In this section we make further constructions corresponding to a given l-planed family of hypersurfaces (Γ,X,P ) over B. These will formalize the techniques of Proposition 2.3 and Example 3.1. We assume that the schemes Π,Y,Z, etc., of Section 2.3.4 correspond to our given l-planed family. Let G := Grassk(Γ) be the family of k-planes contained in the projective bundle Γ. There is a map Grassk(ΓΠ) → G coming from the map of B-schemes ∼ (12) Grassk(ΓΠ) −→ Grassk(Γ) ×B Π −→ Grassk(Γ) = G, from which we get another map

ψ : Fk(Z/Π) ,→ Grassk(ΓΠ) −→ G.

Now, each fiber of Fk(Z/Π) → G over a k-plane Λb ∈ Gb is in fact a linear subspace of ∼ n−l−1 Πb = P . This follows form Proposition 2.9 of Section 2.4: each point of Γb simply imposes a linear condition on the linear series {ZΘb }Θb∈Πb , so the points of Λb merely impose so many linear conditions on the parameters Θb ∈ Πb. (By applying the definition of the relative Fano variety to the discussion of Section 2.4, one is readily convinced that these linear conditions cut out the fiber in Fk(Z/Π) over Λb scheme-theoretically, so that the fiber is a reduced linear space.) However, we cannot be assured that all of the fibers have the same dimension; although generically, they do, by the Theorem on Fiber Dimension (1.3). Let C be the open subset of the irreducible projective variety G over which the fibers of this map ψ have the minimum k+d−1 possible dimension l − k , i. e. the set of k-planes imposing independent conditions on the linear series. At present we do not know that this is nonempty; this will follow from Corollary 3.12, if the dimension l is appropriately high. Thus, from Lemma 3.3, the restriction to C of Fk(Z/Π) → G is in fact a projective bundle over C, which we will denote by Q :

Fk(Z/Π)C =: Q −→ C ⊂ G. We also have a morphism Q −→ Π, since Q is a sub-bundle of the restriction to C of the n−l−1 P -bundle G ×B Π −→ G. A point of the variety Q over a closed point b ∈ B can be thought of as a pair (Ωb, Θb) ∈

Grassk(Γb) × Πb, such that such that Ωb ⊂ ZΘb —in other words,

Ωb ⊂ ZΘb ⊂ Γb ⊂ Θb ⊂ Pb, which also implies that

Ωb ⊂ YΘb ⊂ Θb 32 AUTHOR: ALEX WALDRON ADVISOR: JOE HARRIS for a point (Ωb, Θb) ∈ Q. If G˜ → G is the universal family, then this last inclusion becomes

(13) G˜C ⊂ Q.

On the other hand, we let Π0 ⊂ DEF(Z) ⊂ Π be the maximal open subset over which the ﬁbers of Z → Π have the dimension l − 1 and are smooth. In other words,

Π0 ⊂ DEF(Z) ⊂ Π is the set of Θb such that YΘb does not contain the entire l-plane Γb, and meets Γb in a smooth hypersurface ZΘb . Finally, let D ⊂ Q be the inverse image of Π0 in Q. Thus, the ﬁber over a closed point d = (Ωb, Θb) ∈ D of the family ZD → D is the smooth hypersurface ZΘb = Zd of dimension l − 1 in Γb, and Ωb ⊂ Zd. In the following diagram of B-schemes, we summarize the constructions we have made from the given family (Γ,X,P ). Here, the symbol “⊂” indicates an open embedding, and “,→” indicates a closed embedding.

(14) Z / Y m D K D D mmm KK | mm T KK | mmm KK || mm KK || mmm KK ~ | mv mm % | G˜ / C o Q Π0 Z /n Y D JJ _ N JJ ~~ }} T T JJ T ~ } JJ ~~ }} JJ ~~ }} JJ$ ~~ }~ } G˜ / G = Grassk(Γ) o Fk(Z/Π) / Π o Π˜ h t: QQQ _ tt QQ tt QQQ tt QQ tt QQ tt Grassk(ΓΠ) The following is the purpose of these constructions:

Proposition 3.11. The triple (G˜D,YD, Π˜ D) is a k-planed family of hypersurfaces of degree d − 1 over D. The bundle G˜D is contained in the smooth locus of YD → D. Furthermore, if D and C are non-empty then D −→ C −→ B is a comb over B. Note. For this to be nontrivial, the open sets C ⊂ G and D ⊂ Q must be nonempty: this will be shown in Section 3.4.2 under the appropriate circumstances.

Proof. The inclusion G˜D ⊂ YD is the pullback of equation (13). The family YD was originally deﬁned as a subscheme of Π˜ D in Section 2.3.4.

For the second statement, let d = (Ωb, Θb) ∈ D be a closed point. Recall that Zd = ZΘb is by deﬁnition smooth, and Ωb ⊂ ZΘb = Yb ∩Γb. Therefore Ωb ⊂ (Yd)sm, which by Remark 2.10 is equivalent to the desired statement. For the third statement, D ⊂ Q −→ C ⊂ G −→ B is by deﬁnition a comb morphism, provided D and C are non-empty. FANO VARIETIES AND UNIRATIONALITY 33

3.4. Proof of unirationality of smooth low-degree hypersurfaces. In Section 3.4.2, we will show that for a family of hypersurfaces X → B of appropriately high dimension relative to its degree d (see section 3.2), the natural map YD → X is dominant. The degree of the family of hypersurfaces YD is equal to d − 1, which we will use to induct on degree in Section 3.4.3.

3.4.1. Rough description of the proof. In the induction argument to follow, the location of the parametrizing variety itself is obscured. It is therefore instructive to give the following rough description of the proof of Theorem 3.6: Apply Proposition 3.11 to obtain successive families of hypersurfaces Y d−i of degree Dd−i d − 1, d − 2, d − 3..., with each family dominating the previous ones; their bases Dd−i will form a comb over the original base B, provided the Di can be chosen nonempty (we will establish this using the dimension, degree, and smoothness requirements of the theorem). This can be drawn out as follows: o Y d−1 o Y d−2 o ... o Y 1 = D X Dd−1 Dd−2 D1 0

B o Dd−1 o Dd−2 o ... o D1. We will arrive at a family of hypersurfaces Y 1 of degree d = 1 dominating all prior D1 families Y d−i . By Proposition 3.3, this last family Y 1 is then itself a projective bundle Dd−i D1 over D1, which we may call D0. Hence D0 → D1 → ... → B will be a comb over B (if non-empty), with a dominant map D0 → X given by the composition of dominant maps D → Y 2 → · · · → X. 0 D2 To obtain Corollary 3.7, simply observe that if the original base B is rational, then the bases Di at each step are rational (see Proposition 3.5). This includes the last family D0, which is rational and will dominate all the previous varieties Y i and X. Hence X will be Di unirational. In case B = Spec(K) and X is a single hypersurface, the parametrizing variety is r a certain subvariety of a cross-product of ﬂag manifolds in P (see [2] for an explicit description).

3.4.2. Dominance of YD → X. We proceed to establish the remaining facts which are necessary to show that YD → X is dominant. These are obtained directly from Theorem 2.13 of the previous chapter. Corollary 3.12. Let d and k be positive integers, and let N = N(d, k) as deﬁned in Section 2.5. Let l ≥ N(d, k) be an integer, and let l m D = {Dα ⊂ P }α∈P l m be a base-point-free linear series of hypersurfaces of degree d in P parametrized by P (necessarily, m ≥ l). Let m Ψ = {(α, Λ) ∈ P × G(k, l) | Λ ⊂ Dα} be the incidence correspondence. Then, 34 AUTHOR: ALEX WALDRON ADVISOR: JOE HARRIS

l (a) A general k-plane Ω ⊂ P imposes independent conditions on the linear series D, that is, k + d dim({α ∈ m | Ω ⊂ D }) = m − ≥ 0. P α d (b) The variety Ψ has dimension dim(Ψ) = m + φ(n, d, k), m and every irreducible component of Ψ projects surjectively onto the ﬁrst factor P . There is a unique irreducible component of Ψ dominating G(k, l), and this in particular projects m surjectively onto P . Proof. This is derived by combining Lemma 2.17 and Theorem 2.13. k+d (a) Note that m ≥ l ≥ N(d, k) ≥ d , which is the maximum possible number of conditions imposed by a k-plane. So π2 :Ψ → G(k, l) is surjective, with general ﬁber of dimension k + d dim(Ψ) − dim( (k, l)) = m + φ(l, d, k) − (k + 1)(l − k) = m − . G d

m (b) Consider the projection map π1 :Ψ → P . From Lemma 2.17, we know that a general member Dα of the linear series is smooth; so from Theorem 2.13, the general fiber of π1 has dimension dim(Fk(Dα)) = φ(l, d, k) ≥ 0, and in particular π1 is surjective (i. e. all hypersurfaces Dα contain k-planes. As in the proof of Theorem 2.17, set o m Sh = {α ∈ P | dim((Dα)sing)) = h}. Then for each h = 0, . . . , l − 1, we have o dim(Sh) ≤ m − h − 1. o So by Corollary 2.14(b), for any α ∈ Sh, −1 dim(π1 (α)) < φ(l, d, k) + h + 1 o m ≤ φ(l, d, k) + codim(Sh ⊂ P ), from which we may write −1 o o −1 dim(π1 (Sh)) ≤ dim(Sh) + dim(π (Λ)) < m + φ(l, d, k). d+k On the other hand, Ψ is cut out by k conditions, so for each component Ψ0 of Ψ, d + k dim(Ψ ) ≥ m + dim( (k, l)) − = m + φ(l, d, k). 0 G k o Thus π1(Ψ0) 6⊂ Sh for any h, and the minimum fiber dimension over π1(Ψ0) is φ. Corollary m 1.4(b) therefore implies that dim(Ψ0) = m+φ−φ = m and ψ0 surjects onto P , as desired. (Here we have “inflated” the image of Ψ0 by bounding the dimension of the fibers, which is the contribution of Theorem 2.13.) We have from (a) that Ψ surjects onto G(k, l). If two components Ψ1, Ψ2 both dominated, then the open sets Ψ1 − Ψ2 and Ψ2 − Ψ1 also dominate; so the generic fiber would FANO VARIETIES AND UNIRATIONALITY 35 be reducible. But each set-theoretic fiber over G(k, l) is a projective space, so a unique component of Ψ dominates. Now, assume given an l-planed family of hypersurfaces (Γ,X,P ) over an integral base B, and assume that the constructions of section 3.3 have been made for this family. Proposition 3.13. If X → B is smooth along Γ and l ≥ N(d−1, k), then D is nonempty, hence a comb over B, and the natural B-morphism YD → X is fiber-by-fiber dominant over B (and therefore dominant). Proof. (It may be helpful here to refer to the diagram (14).) As before, it will suffice to work over a single closed point b ∈ B, so we assume temporarily that B = Spec(K). Thus ∼ l X = Xb is smooth along Γ = Γb = P , and by Proposition 2.9(b), the secondary residual schemes {ZΘ} form a base-point free linear series of degree d − 1 in Γ (a basepoint would be singular). There is a trivial technicality coming from Proposition 2.9: the linear series {ZΘ} is r−l−1 m parametrized not by Π = P but by a quotient of Π, which we will call P . There is thus m p a surjective map Π ⊃ DEF(Z) P whose fibers are projective subspaces P ⊂ DEF(Z) of rank p = r − l − m − 1. This map induces a surjective rational map α from the Fano variety (over b), n−l−1 Fk(Z/Π) ⊂ (Grassk(Γ) ×B Π)b = G(k, l) × P , m onto the the incidence correspondence “Ψ”⊂ G(k, l) × P of Corollary 3.12: α Fk(Z/Π) ____ / / Ψ u @ II uu @ II uu @@ uIuI @@ uu I @ uz u II @@ I$ m G = G(k, l) Π ___ / / P . p Each fiber of this map α is of course also a projective space P . k+d−1 Now, by Corollary 3.12(a), since l ≥ N(d−1, k) ≥ k , the general k-plane imposes independent conditions on the linear series {ZΘ}, and each fiber is non-empty. Thus Ψ → G is surjective, as is Fk(Z/Π) → G, and the open subset C ⊂ G is non-empty. So, the projective bundle Q ⊂ Fk(Z/Π) is a non-empty open subset. On the other hand, the open subset Π0 ⊂ Π is also non-empty (over b): for, DEF(Z)b is non-empty by Corollary 3.12(b), and the general ZΘ is smooth by Lemma 2.17. So D ⊂ Q, the inverse image of Π0. Therefore, following Proposition 3.11, D → C → B is in fact a comb. We now show that YD → X dominates over b. By Corollary 3.12(b), “Ψ” dominates m r−l−1 P , hence Fk(Z/Π) dominates Π = P as do the open subsets Q and D. Let Π1 ⊂ Π0 be an open subset contained in the image of D. Let X0 ⊂ X be the open subset lying outside of both Γ and the inverse image of Π − Π1 under the projection πΓ : X − Γ → Π; this set X0 is non-empty since Π1 ⊂ Π0 ⊂ DEF(Z) ⊂ DEF(Y ). Furthermore, X0 is dense, since X = Xb must be integral in order not to meet Γ in a singular point.

Let Θx = x, Γ.Choose a k-plane Ωx ⊂ ZΘx , so (Ωx, Θx) is a closed point of D (possible since l ≥ N(d − 1, k)). Then because x 6∈ Γ, we have x ∈ Y(Ωx,Θx). Thus x is in the image of Y(Ωx,Θx), and we are done since x ∈ X0 was general. 36 AUTHOR: ALEX WALDRON ADVISOR: JOE HARRIS

3.4.3. Proof of Theorem 3.6. We argue by induction on the degree d. In case d = 1, the family of hypersurfaces is itself a comb. Now, let d > 1 and assume that the conclusions of the theorem hold for all degrees less than d. We must show that the theorem holds for a family (Γ,X,P ) of l-planed hypersurfaces of degree d, with Γ a projective bundle of rank l ≥ L(d) contained in the smooth locus of X → B. Set k = L(d − 1), and let (Ω˜ D,YD, Π˜ D) be the k-planed family of hypersurfaces of degree d − 1 corresponding to (Γ,X,P ), from Section 3.3. By Proposition 3.11, and since l ≥ L(d) = N(d − 1,L(d − 1)) = N(d − 1, k), the family (Ω˜ D,YD, Π˜ D) satisfies the hypotheses of Proposition 3.13. Therefore D → B is a comb, and YD → X is fiber-by-fiber dominant over B. Also by Proposition 3.11 and the definition k = L(d − 1), the family (Ω˜ D,YD, Π˜ D) satisfies the induction hypothesis. Therefore we get a comb F → D and a morphism F → YD which is dominant fiber-by-fiber over D and hence also over B. The composition of comb morphisms F → D → B is again a comb morphism, and the composition

F → YD → X is again a fiber-by-fiber dominant B-morphism. So F is the required comb over B dominating X fiber-by-fiber. This completes the proof of Theorem 3.6. Acknowledgements. My utmost thanks are due to Joe Harris for his vast wisdom and tolerance. (And of course for the results of the last 2/3 of the paper!) Several of my most esteemed colleagues also deserve thanks and recognition: Silas, Luca, Jae and Virginia have been an indispensible source of last-minute insights through the years. This thesis is mostly the work of Emily Fifer, who put up with it with a smile. My mom, dad and grandparents deserve thanks for reasons both obvious and not. The writing of this paper was supported in part by the Harvard College Research Program and PRISE. FANO VARIETIES AND UNIRATIONALITY 37

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